e as a limit

WEBVTT
Kind: captions
Language: en

00:00:00.503 --> 00:00:01.752
Narrator: In a previous video

00:00:01.752 --> 00:00:03.403
when we were looking at a very simple case

00:00:03.403 --> 00:00:06.143
of compounding interest,
we got the expression

00:00:06.143 --> 00:00:14.541
(1+1/n)^n and the way we got this,

00:00:14.541 --> 00:00:17.520
we saw an example where
a loan shark is charging

00:00:17.520 --> 00:00:20.755
100% interest and that's where this 1 is,

00:00:20.755 --> 00:00:23.272
and then if they only
compound once in the year,

00:00:23.272 --> 00:00:26.066
so it's 100% over the year, then n is 1.

00:00:26.066 --> 00:00:30.019
So, you get 1+100%/1^1,

00:00:30.019 --> 00:00:31.019
you're going to have to pay back

00:00:31.019 --> 00:00:32.778
twice the amount of the
original amount of money.

00:00:33.102 --> 00:00:39.689
If n is 2, (1+1/2)^2, gets you 2.25

00:00:39.689 --> 00:00:43.852
If you compound half the interest,

00:00:43.852 --> 00:00:47.803
so 100%/2, but you compound it twice.

00:00:47.803 --> 00:00:49.672
Then we kept going and going and going,

00:00:49.672 --> 00:00:51.681
we saw interesting things happen.

00:00:51.681 --> 00:00:53.521
I want to review that right over here

00:00:53.521 --> 00:00:54.516
using this calculator.

00:00:54.516 --> 00:00:57.082
I want to see what
happens as we get larger

00:00:57.082 --> 00:00:58.269
and larger and larger n's.

00:00:58.269 --> 00:01:01.476
In that last video we
went as high as n=365

00:01:01.476 --> 00:01:04.354
and it seemed to be
approaching a magical number,

00:01:04.354 --> 00:01:06.254
but now let's go even further.

00:01:06.608 --> 00:01:12.871
So, let's type in ... let's throw
some really large numbers here.

00:01:12.871 --> 00:01:23.072
1+1/1,000,000 so that's a
million to the millionth power.

00:01:23.349 --> 00:01:27.242
(1+1/1,000,000)^1,000,000

00:01:27.489 --> 00:01:28.434
Did I get the right number of zero's?

00:01:28.434 --> 00:01:29.353
Yeah, that looks right.

00:01:29.353 --> 00:01:32.293
Before I even press
enter, which is exciting,

00:01:32.293 --> 00:01:33.872
let's just think about
what's going on here.

00:01:33.872 --> 00:01:35.719
This part that we have here is that

00:01:35.719 --> 00:01:37.562
n gets larger and larger,
it's getting closer

00:01:37.562 --> 00:01:40.015
and closer to 1, but
never quite exactly 1.

00:01:40.015 --> 00:01:42.027
This is 1 and 1 millionth.

00:01:42.027 --> 00:01:44.433
So, it's very close to
1, but not exactly 1.

00:01:44.433 --> 00:01:46.936
We're going to raise that
thing to the millionth power

00:01:46.936 --> 00:01:48.023
and normally when you raise something

00:01:48.023 --> 00:01:49.267
to the millionth power,
that's just going to

00:01:49.267 --> 00:01:52.020
be unbounded, just
become some huge number,

00:01:52.020 --> 00:01:55.870
but there's a clue that 1 to the
millionth power will just be 1.

00:01:55.870 --> 00:01:57.266
If we're getting really close to 1,

00:01:57.266 --> 00:01:59.929
well maybe this won't just
be some unbounded number.

00:02:00.636 --> 00:02:03.203
When we calculate it, we
see that that's the case.

00:02:03.203 --> 00:02:06.272
It's 2.71828 and just keeps going.

00:02:06.442 --> 00:02:07.536
Now, let's go even higher.

00:02:07.536 --> 00:02:17.019
Let's take it ... let's
do 1+1/ and actually

00:02:17.019 --> 00:02:20.136
I can now use scientific notation.

00:02:20.490 --> 00:02:32.675
Let's just say (1+1/1'10^7)^1'10^7,

00:02:32.675 --> 00:02:33.637
so what do we get here?

00:02:34.099 --> 00:02:38.810
So, now we went 2.718281692.

00:02:38.810 --> 00:02:40.139
Let's go even larger.

00:02:40.769 --> 00:02:41.934
Let's get our last entry here.

00:02:41.934 --> 00:02:44.341
Let's go, instead of the 7th power,

00:02:44.341 --> 00:02:45.635
let's go to the eighth power,

00:02:45.635 --> 00:02:51.068
so now we're (1+1/100,000,000^100,000,000)

00:02:51.068 --> 00:02:53.270
I don't even know if this
calculator can handle this

00:02:53.270 --> 00:02:59.633
and we get 2.71828181487

00:02:59.633 --> 00:03:02.076
and you see that we are
quickly approaching,

00:03:02.076 --> 00:03:03.803
or maybe not so quickly,
we have to raise this

00:03:03.803 --> 00:03:07.036
to a very large power, to the number e.

00:03:07.375 --> 00:03:08.803
The number e in our calculator.

00:03:08.803 --> 00:03:14.807
You see we've already gotten
1, 2, 3, 4, 5, 6, 7 digits

00:03:14.807 --> 00:03:16.613
to the right of the decimal point

00:03:16.613 --> 00:03:19.533
by taking it to the 100 millionth power.

00:03:19.825 --> 00:03:21.542
So, we are approaching this number.

00:03:22.172 --> 00:03:25.554
We are approaching, so
one way to talk about it

00:03:25.554 --> 00:03:32.400
is we could say the limit,
as n approaches infinity.

00:03:32.400 --> 00:03:35.743
As n becomes larger and larger,
it's not becoming unbounded.

00:03:35.743 --> 00:03:37.511
It's not going to infinity.

00:03:37.511 --> 00:03:39.845
It seems to be approaching this number

00:03:39.845 --> 00:03:41.936
and we will call this number,

00:03:41.936 --> 00:03:46.612
we will call this magical
and mystical number e.

00:03:47.104 --> 00:03:48.734
We'll call this number e and we see

00:03:48.734 --> 00:03:51.458
from our calculator that this number

00:03:51.458 --> 00:03:53.599
and these are kind of, these are almost

00:03:53.599 --> 00:03:56.269
as famous digits as the digits for Pi,

00:03:56.715 --> 00:04:04.538
we are getting 2.7182818 and it just

00:04:04.538 --> 00:04:07.267
keeps going and going and going.

00:04:07.267 --> 00:04:09.809
Never, never repeating,
so it's an infinite

00:04:09.809 --> 00:04:12.215
string of digits, never, never repeating.

00:04:12.215 --> 00:04:14.603
Just like Pi. Pi, you
remember, is the ratio

00:04:14.603 --> 00:04:17.101
of the circumference to
the diameter of the circle.

00:04:17.101 --> 00:04:19.351
e is another one of these crazy numbers

00:04:19.351 --> 00:04:20.868
that shows up in the universe.

00:04:20.868 --> 00:04:22.869
And in other videos on Khan Academy

00:04:22.869 --> 00:04:25.536
we go into depth, why this
is so magical and mystical.

00:04:25.536 --> 00:04:26.874
Already this is kind of cool.

00:04:26.874 --> 00:04:28.604
That I can take an infinite ...

00:04:29.034 --> 00:04:33.536
If I just add 1 over a number to 1

00:04:33.536 --> 00:04:35.417
and take it to that number
and I make that number

00:04:35.417 --> 00:04:37.667
larger and larger and
larger, it's approaching

00:04:37.667 --> 00:04:40.753
this number, but what's
even crazier about it

00:04:40.753 --> 00:04:42.203
is we'll see that this number,

00:04:42.203 --> 00:04:44.019
which you can view, one way of it,

00:04:44.019 --> 00:04:46.937
it is coming out of
this compound interest.

00:04:47.613 --> 00:04:50.538
That number, Pi, the imaginary unit

00:04:50.538 --> 00:04:53.485
which is defined as that
imaginary unit squared

00:04:53.485 --> 00:04:54.535
is a negative 1,

00:04:54.535 --> 00:04:56.337
that they all fit together in this magical

00:04:56.337 --> 00:04:57.134
and mystical way

00:04:57.134 --> 00:04:59.096
and we'll see that again in future videos.

00:04:59.204 --> 00:05:00.736
But just for the sake of e,

00:05:00.736 --> 00:05:02.760
what you could imagine
what's happening here

00:05:02.760 --> 00:05:05.076
is going to our previous example

00:05:05.076 --> 00:05:09.867
of borrowing $1 and trying
to charge 100% over a year,

00:05:09.867 --> 00:05:11.869
when our n was 1, that means you're just

00:05:11.869 --> 00:05:12.873
charging over 1 period.

00:05:12.873 --> 00:05:14.733
When n is 2, you're
charging over 2 periods

00:05:14.733 --> 00:05:17.808
and then compounding, or you're
compounding over 2 periods.

00:05:17.808 --> 00:05:20.207
When n is 3, you're
compounding over 3 periods.

00:05:20.207 --> 00:05:22.086
When n approaches infinity,

00:05:22.086 --> 00:05:24.769
you could view it as you're
continuously compounding

00:05:24.769 --> 00:05:26.605
every zillionth of a second.

00:05:26.605 --> 00:05:30.001
Every moment you're compounding

00:05:30.001 --> 00:05:32.886
in a super small amount of interest,

00:05:32.886 --> 00:05:35.518
but you're doing it,
essentially you're approaching

00:05:35.518 --> 00:05:36.802
an infinite number of times

00:05:36.802 --> 00:05:38.048
and you get to this number.


e and compound interest

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:03.935
Let's say that you are
desperate for a dollar.

00:00:03.935 --> 00:00:06.763
So you come to me the local loan shark,

00:00:06.763 --> 00:00:08.805
and you say hey I need
to borrow a dollar for

00:00:08.805 --> 00:00:11.603
a year. I tell you I'm
in a good mood, I willing

00:00:11.603 --> 00:00:14.516
to lend you that dollar
that you need for a

00:00:14.516 --> 00:00:17.008
year. I will lend it to
you for the low interest

00:00:17.008 --> 00:00:25.734
of 100% per year. 100% per year.

00:00:25.734 --> 00:00:28.348
How much would you have
to pay me in a year?

00:00:28.348 --> 00:00:30.203
You're going to have to pay the original

00:00:30.203 --> 00:00:33.941
principal what I lent
you plus 100% of that.

00:00:33.941 --> 00:00:37.435
Plus one other dollar.
Which is clearly going

00:00:37.435 --> 00:00:44.349
to be equal to $2. You
say oh gee, that's a lot

00:00:44.349 --> 00:00:48.468
to have to pay to pay back
twice what I borrowed.

00:00:48.468 --> 00:00:50.942
There's a possibility that
I might have the money

00:00:50.942 --> 00:00:54.537
in 6 months. What kind
of a deal could you get

00:00:54.537 --> 00:00:56.470
me for that Mr. Loanshark?

00:00:56.470 --> 00:00:58.742
I say oh gee, if your
willing to pay back in

00:00:58.742 --> 00:01:01.074
6 months, then I'll
just charge you half the

00:01:01.074 --> 00:01:03.146
interest for half the time.

00:01:03.146 --> 00:01:10.284
You borrow one dollar,
so in 6 months, I will

00:01:10.284 --> 00:01:18.111
charge you 50% interest.
50% interest over 6 months.

00:01:18.111 --> 00:01:22.691
This, of course, was 1
year. How much would you

00:01:22.691 --> 00:01:24.868
have to pay? Well, you
would have to pay the

00:01:24.868 --> 00:01:26.872
original principal what you borrowed.

00:01:26.872 --> 00:01:29.601
The one dollar plus
50% of that one dollar.

00:01:29.601 --> 00:01:35.299
Plus 0.50, and that of
course is equal to 1.5.

00:01:36.037 --> 00:01:39.923
That is equal to $1.50. I'll
just write it like this.

00:01:39.923 --> 00:01:44.087
$1.50. Now you say well
gee that's I guess better.

00:01:44.087 --> 00:01:46.524
What happens if I don't
have the money then?

00:01:46.524 --> 00:01:49.060
If I still actually need a year.

00:01:49.060 --> 00:01:51.667
We actually have a system for that.

00:01:51.667 --> 00:01:53.858
What I'll do is just
say that okay, you don't

00:01:53.858 --> 00:01:56.004
have the money for me
yet. I'll essentially ...

00:01:56.004 --> 00:01:57.756
we could think about it.
I will just lend that

00:01:57.756 --> 00:02:00.937
amount that you need for
you for another 6 months.

00:02:00.937 --> 00:02:02.751
We'll lend that out.

00:02:02.751 --> 00:02:05.798
We'll lend that out for
another 6 months at the

00:02:05.798 --> 00:02:10.837
same interest rate at 50%
for the next 6 months.

00:02:10.837 --> 00:02:17.324
Then you'll owe me the
principal a $1.50 plus

00:02:17.324 --> 00:02:24.314
50% of the principal, plus 75
cents. Plus 75 cents, and that

00:02:24.314 --> 00:02:30.517
gets us to $2.25. That equals $2.25.

00:02:30.517 --> 00:02:33.295
Another way of thinking
about it is to go from

00:02:33.295 --> 00:02:35.514
$1 over the first
period, you just multiply

00:02:35.514 --> 00:02:38.715
that times 1.5. If your
going to grow something

00:02:38.715 --> 00:02:41.747
by 50%, you just multiply it times 1.5.

00:02:41.747 --> 00:02:43.807
If your going to grow it
by another 50%, you can

00:02:43.807 --> 00:02:46.127
multiply by 1.5 again.

00:02:46.127 --> 00:02:48.400
One way of thinking about
it that 50% interest is

00:02:48.400 --> 00:02:51.271
the same thing as multiplying by 1.5.

00:02:51.271 --> 00:02:53.397
Multiplying by 1.5.

00:02:53.397 --> 00:02:58.448
If you start with 1 and
multiply by 1.5 twice,

00:02:58.448 --> 00:03:00.804
this is going to be the same thing.

00:03:00.804 --> 00:03:06.399
$2.25 is going to be 1
multiplied by 1.5 twice.

00:03:06.399 --> 00:03:12.021
1.5 multiplied twice is the
same thing as 1.5 squared.

00:03:12.021 --> 00:03:13.801
You can see the same
thing right over here.

00:03:13.801 --> 00:03:17.281
This is the same thing.
100% is the same thing

00:03:17.281 --> 00:03:22.997
as multiplying by 2. As we
be multiplying 1 plus 1.

00:03:22.997 --> 00:03:26.025
This is multiplying by
2, so you could do this

00:03:26.025 --> 00:03:31.694
right over here. You could
do this as 1 times 1.

00:03:31.694 --> 00:03:41.849
1 times 1 to the first ... I'm sorry
1 times 2 to the first power , because

00:03:41.849 --> 00:03:44.180
your only doing it over
one period over that year.

00:03:44.180 --> 00:03:46.248
You say once again
where's that 2? Well, if

00:03:46.248 --> 00:03:48.785
someone is asking for
100%, that means over the

00:03:48.785 --> 00:03:50.580
period you're going have to pay twice.

00:03:50.580 --> 00:03:52.320
You're going have to pay
the principal plus 100%.

00:03:52.320 --> 00:03:54.915
You're going have to pay
twice what you originally

00:03:54.915 --> 00:03:57.069
borrowed. If someone is
charging you 50% over

00:03:57.069 --> 00:03:59.520
every period, you're
going have to pay whatever

00:03:59.520 --> 00:04:02.246
you borrowed. That's
kind of the one part plus

00:04:02.246 --> 00:04:05.514
50% of it. So 1.5 times what
you borrowed. You multiply

00:04:05.514 --> 00:04:09.906
times 1.5 every time.
If you wanted to see how

00:04:09.906 --> 00:04:12.311
this actually related to
the interest, you could

00:04:12.311 --> 00:04:14.790
view this as ... this
right over here is equal

00:04:14.790 --> 00:04:22.736
to 1 times, the interest
part is 1 plus 100%

00:04:22.736 --> 00:04:26.571
divided by 1 period to the first power.

00:04:26.571 --> 00:04:28.815
I know this seems like
a crazy way of rewriting

00:04:28.815 --> 00:04:31.391
what we just wrote over
here. Writing 1 plus 1,

00:04:31.391 --> 00:04:33.855
but you'll see that we
can keep writing this

00:04:33.855 --> 00:04:36.185
as we compound over different periods.

00:04:36.185 --> 00:04:38.737
This one right over here, we can rewrite.

00:04:38.737 --> 00:04:44.993
We can write as 1 times 1 plus 100%.

00:04:44.993 --> 00:04:47.736
Here we took our 100% for the year, and we

00:04:47.736 --> 00:04:50.403
divided into 2 periods.
Two 6 month periods.

00:04:50.403 --> 00:04:54.094
Each of them at 50%. 1
plus 100% over 2 is the

00:04:54.094 --> 00:04:58.801
same thing as 1.5, and
we compounded it over

00:04:58.801 --> 00:05:01.121
2 periods. Let me do that
2 periods into a different

00:05:01.121 --> 00:05:05.516
color. The periods, let
me do in this orange

00:05:05.516 --> 00:05:08.525
color right over here.
You might start to see a

00:05:08.525 --> 00:05:10.604
pattern forming. Let's
say, well gee, I might

00:05:10.604 --> 00:05:13.017
have the money back in
... and you don't really

00:05:13.017 --> 00:05:15.326
like this. This is
$2.25. That was more than

00:05:15.326 --> 00:05:17.934
the original $2, so you
say, well what if we do

00:05:17.934 --> 00:05:20.400
this over every 12
months. I say, "Sure. We

00:05:20.400 --> 00:05:22.126
got a program for that."

00:05:22.126 --> 00:05:26.432
After every 12 months
... or after every month

00:05:26.432 --> 00:05:31.355
I should say, I'm just
going to charge you 100%

00:05:31.355 --> 00:05:38.525
divided by 12 interest.
This is equal to 8 1/3%.

00:05:38.525 --> 00:05:44.731
Having to pay back the
principal plus 8 1/3%,

00:05:44.731 --> 00:05:50.463
that's the same thing as
multiplying times 1.083 repeating.

00:05:50.463 --> 00:05:55.658
After 1 month you would
have to pay 1.083 repeating.

00:05:55.658 --> 00:05:59.589
After 2 months ... and
this isn't the scale

00:05:59.589 --> 00:06:02.188
that actually looks
more than 2 months, but

00:06:02.188 --> 00:06:04.462
it's not completely at
scale. After 2 months

00:06:04.462 --> 00:06:06.792
your going have to multiply by this again.

00:06:06.792 --> 00:06:11.689
Times 1.083 repeating,
and so that would get you

00:06:11.689 --> 00:06:18.066
1.083 repeating squared.
If you went all the way

00:06:18.066 --> 00:06:20.614
down 12 months ... let
me get myself some space

00:06:20.614 --> 00:06:24.869
here. If you went all the way
down 12 months ... let me just.

00:06:27.607 --> 00:06:30.043
I should way from the
beginning 12 months, so

00:06:30.043 --> 00:06:32.429
another 10 months.
What's the total interest

00:06:32.429 --> 00:06:34.397
you would have to pay
over a year if you weren't

00:06:34.397 --> 00:06:35.866
able to keep coming up with the money?

00:06:35.866 --> 00:06:37.399
If you had to keep re-borrowing it.

00:06:37.399 --> 00:06:39.340
I kept compounding that interest.

00:06:39.340 --> 00:06:44.436
Well, you're going have
to pay 1.083 to the ...

00:06:44.436 --> 00:06:47.932
this is for 1 month. You
could view this as to

00:06:47.932 --> 00:06:50.858
the first power. This is
for 2 months, so you're

00:06:50.858 --> 00:06:53.202
going have to pay this to the 12th power.

00:06:53.202 --> 00:06:56.667
We have compounded over
12 periods, 8 1/3% over

00:06:56.667 --> 00:06:58.939
12 periods. If you wanted
to write it in this

00:06:58.939 --> 00:07:01.472
form right over here,
this would be the same

00:07:01.472 --> 00:07:03.470
thing as the original principal.

00:07:03.470 --> 00:07:12.028
Our original principal
times 1 plus 100% divided

00:07:12.028 --> 00:07:17.136
by 12. Now we've divided
our 100% into 12 periods,

00:07:17.136 --> 00:07:20.604
and we're going to compound that 12 times.

00:07:20.604 --> 00:07:26.665
We're going to take
that to the 12th power.

00:07:26.665 --> 00:07:28.739
What is this going to
equal to? This buisness

00:07:28.739 --> 00:07:31.823
over here. We can get a
calculator out for that.

00:07:33.054 --> 00:07:36.015
I'll get my TI-85 out.
What is this going to be

00:07:36.015 --> 00:07:38.535
equal to? We could do it a couple of ways.

00:07:38.535 --> 00:07:43.869
This is 1.083 repeating.
Let's get our calculator

00:07:43.869 --> 00:07:47.123
out. We could do it a couple of
ways. Let me write it this way.

00:07:47.123 --> 00:07:50.128
Your going to get the same value.
I don't have to rewrite this one.

00:07:50.128 --> 00:07:52.460
I just did that there to
kind of hopefully you'd

00:07:52.460 --> 00:07:54.756
see the kind of structure
in this expression.

00:07:54.756 --> 00:07:58.347
1 plus ... 100% is the same thing as 1.

00:07:58.347 --> 00:08:09.600
1 divided by 12 to the 12th
power. 2.613, I'll just round.

00:08:09.600 --> 00:08:16.527
So approximately 2.613.
You say well this is an

00:08:16.527 --> 00:08:17.929
interesting game you all most forgot about

00:08:17.929 --> 00:08:19.597
your financial troubles,
and you're just intrigued

00:08:19.597 --> 00:08:21.861
by what happens if we keep going this.

00:08:21.861 --> 00:08:25.101
Here we compounded just
... we have 100% over

00:08:25.101 --> 00:08:27.726
here. Here we do 50%
every 6 months. Here we

00:08:27.726 --> 00:08:32.186
do a 12th of 100%, 8 1/3% every 12 months

00:08:32.186 --> 00:08:33.532
until we get to this number.

00:08:33.532 --> 00:08:35.432
What happens if we did every day?

00:08:35.432 --> 00:08:37.196
Every day.

00:08:37.196 --> 00:08:39.266
If I borrowed a one
dollar, and I'd say well

00:08:39.266 --> 00:08:41.263
gee I'm just going to ... each
day I'm going to charge you

00:08:41.263 --> 00:08:45.683
charge you one three hundred
sixty-fifth of a 100%.

00:08:45.683 --> 00:08:50.861
So, 100% divided 365,
and I'm going to compound

00:08:50.861 --> 00:08:56.013
that 365 times. You're
curious mathematically.

00:08:56.013 --> 00:08:57.332
You say well, what do we get then?

00:08:57.332 --> 00:09:01.132
What do we get after a year?

00:09:01.132 --> 00:09:03.595
You have your original
principal. Let me scroll

00:09:03.595 --> 00:09:05.931
over a little bit more to the
right, so we have more space.

00:09:05.931 --> 00:09:08.200
You're going to have
your original principal

00:09:08.200 --> 00:09:14.933
times 1 plus 100% divided
by not 12. Now we've

00:09:14.933 --> 00:09:18.095
divided the 100% into 365 periods.

00:09:18.095 --> 00:09:22.869
365 periods. We're going to compound it.

00:09:22.869 --> 00:09:26.197
Every time we have to
multiply by 1 plus 100%

00:09:26.197 --> 00:09:29.532
over 365 everyday that
the loan is not paid.

00:09:29.532 --> 00:09:33.663
365th power.

00:09:33.663 --> 00:09:35.797
You say oh gee taking
somebody to 365th power

00:09:35.797 --> 00:09:38.599
that's going to give me some huge number.

00:09:38.599 --> 00:09:40.869
Then you say well maybe
not so bad, because

00:09:40.869 --> 00:09:44.012
100% divided by 365 is
going to be a small number.

00:09:44.012 --> 00:09:46.062
This thing is going to
be reasonable close to 1.

00:09:46.062 --> 00:09:48.197
Obviously, we can raise
1 to whatever power we

00:09:48.197 --> 00:09:49.863
want, and we don't get anything crazy.

00:09:49.863 --> 00:09:51.596
Let's see where this one goes.

00:09:51.596 --> 00:09:53.865
Let's see where this one goes.

00:09:53.865 --> 00:09:57.596
This is the same thing
as 1 plus. 100% is the

00:09:57.596 --> 00:10:05.692
same thing as 1 divided
by 365 to the 365th power.

00:10:05.692 --> 00:10:15.710
We get 2.71456. Let me put
it over here. Then we get ...

00:10:15.725 --> 00:10:22.730
This is approximately equal
to ... this approximate

00:10:22.730 --> 00:10:26.901
is a very precise
approximation, but 2.7 ...

00:10:26.901 --> 00:10:30.260
but my calculators
precision only goes so far.

00:10:30.260 --> 00:10:36.827
2.7145675 and it keeps going on and on.

00:10:36.827 --> 00:10:38.860
This is really really
interesting. It looks

00:10:38.860 --> 00:10:41.570
as if we take larger
and larger numbers here,

00:10:41.570 --> 00:10:43.594
it just doesn't just
balloon into some crazy

00:10:43.594 --> 00:10:46.865
ginormous number. It seems
to be approaching some

00:10:46.865 --> 00:10:50.651
magical and mystical
number. It is, in fact,

00:10:50.651 --> 00:10:53.659
the case. That if you
would just take larger and

00:10:53.659 --> 00:10:55.661
larger ... if you were
to take your 100% and

00:10:55.661 --> 00:10:57.324
divide by larger and
larger numbers, but take

00:10:57.324 --> 00:10:59.656
it to that power, you're going to approach

00:10:59.656 --> 00:11:02.721
perhaps the most magical
and mystical number

00:11:02.721 --> 00:11:06.829
of all. The number E.
You can see it right over

00:11:06.829 --> 00:11:09.410
here in your calculator.
They have this E to the X.

00:11:09.410 --> 00:11:12.526
I can do that, so E to
the ... I'll raise it to

00:11:12.526 --> 00:11:14.194
the first power so you can look at the

00:11:14.194 --> 00:11:16.122
calculators internal representation of it.

00:11:16.122 --> 00:11:19.189
You see all ready raising
some ... doing 1 plus 1

00:11:19.189 --> 00:11:21.922
over 365 to the 365th power, we got

00:11:21.922 --> 00:11:24.826
pretty ... we're starting
to get really really

00:11:24.826 --> 00:11:27.997
close to E. I encourage
you try this with larger

00:11:27.997 --> 00:11:29.660
and larger numbers, and your going to get

00:11:29.660 --> 00:11:32.414
closer and closer to this magical mystery.

00:11:32.414 --> 00:11:33.997
You almost wouldn't mind
paying the loan shark

00:11:33.997 --> 00:11:36.527
E dollars, because it's
such a beautiful number.


Optimal turns at Indianapolis Motor Speedway with JR Hildebrand

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.290
SAL KHAN: This is Sal here
with famous Indy car driver--

00:00:04.290 --> 00:00:07.070
smiling when I said
famous-- JR Hildebrand.

00:00:07.070 --> 00:00:09.160
And since you're
here, I thought I

00:00:09.160 --> 00:00:11.520
would ask a question that's
always been on my mind.

00:00:11.520 --> 00:00:12.353
JR HILDEBRAND: Yeah.

00:00:12.353 --> 00:00:15.420
SAL KHAN: We have a picture
here of the Indianapolis Motor

00:00:15.420 --> 00:00:17.080
Speedway.

00:00:17.080 --> 00:00:19.869
And I've always
wondered how you--

00:00:19.869 --> 00:00:22.160
it seems like turning is a
very important part of the--

00:00:22.160 --> 00:00:23.620
JR HILDEBRAND: It's
absolutely an important part

00:00:23.620 --> 00:00:24.160
of what we're doing.

00:00:24.160 --> 00:00:24.710
SAL KHAN: --of the race.

00:00:24.710 --> 00:00:25.400
JR HILDEBRAND:
People get fixated

00:00:25.400 --> 00:00:26.530
on the car going straight.

00:00:26.530 --> 00:00:27.960
But the turning part
is pretty important.

00:00:27.960 --> 00:00:29.233
SAL KHAN: Turning seems
to be the part where

00:00:29.233 --> 00:00:30.630
a lot of the skill
comes into it.

00:00:30.630 --> 00:00:33.830
And I've always wondered,
what is optimal?

00:00:33.830 --> 00:00:37.060
Do y'all try to minimize your
distance and kind of take

00:00:37.060 --> 00:00:41.140
the turn as quickly or as in
short of a distance as possible

00:00:41.140 --> 00:00:45.550
by really hugging the
corner, by going like that?

00:00:45.550 --> 00:00:48.310
But when you do that,
you have to turn more.

00:00:48.310 --> 00:00:49.360
There's more g-forces.

00:00:49.360 --> 00:00:51.960
There's more kind
of centripetal force

00:00:51.960 --> 00:00:54.190
that your tires
have to deal with,

00:00:54.190 --> 00:00:55.650
the human has to deal with.

00:00:55.650 --> 00:00:58.480
Versus taking the
outside where you

00:00:58.480 --> 00:01:03.330
have to cover more distance, but
the centripetal acceleration,

00:01:03.330 --> 00:01:06.039
the g-forces aren't
going to be as dramatic.

00:01:06.039 --> 00:01:07.330
So how do you think about that?

00:01:07.330 --> 00:01:08.996
JR HILDEBRAND: Well,
every track ends up

00:01:08.996 --> 00:01:10.450
being a little bit different.

00:01:10.450 --> 00:01:13.276
But when we take Indianapolis
here as the example,

00:01:13.276 --> 00:01:14.650
if you're already
on the inside--

00:01:14.650 --> 00:01:17.240
it's like the 800 meter
runner's kind of path.

00:01:17.240 --> 00:01:18.370
It's the shortest distance.

00:01:18.370 --> 00:01:19.550
You can kind of get
from point A to point

00:01:19.550 --> 00:01:22.290
B. The lap is the same every
time, so it doesn't actually

00:01:22.290 --> 00:01:26.300
depend on you running a
specific distance or not.

00:01:26.300 --> 00:01:30.680
For us, in this example, the
car actually just won't do that.

00:01:30.680 --> 00:01:33.020
If you think about being
all the way on the inside,

00:01:33.020 --> 00:01:34.440
being all the way on the
inside through the corner,

00:01:34.440 --> 00:01:36.231
and then exiting all
the way on the inside,

00:01:36.231 --> 00:01:40.070
it's having to do the most
work to follow that path.

00:01:40.070 --> 00:01:41.810
And in Indianapolis,
we're approaching

00:01:41.810 --> 00:01:45.300
turn one at upwards
of 240 miles per hour.

00:01:45.300 --> 00:01:48.150
And that turn one is
not-- it's hardly banked.

00:01:48.150 --> 00:01:50.000
It looks quite flat in person.

00:01:50.000 --> 00:01:55.750
So as opposed to NASCAR running
at Talladega or Daytona,

00:01:55.750 --> 00:01:58.140
these big, giant
super speedways,

00:01:58.140 --> 00:02:01.010
the car is having to
do quite a lot of work

00:02:01.010 --> 00:02:02.847
to get through the corner here.

00:02:02.847 --> 00:02:03.930
SAL KHAN: So how do you--?

00:02:03.930 --> 00:02:05.194
Do you take the outside or--?

00:02:05.194 --> 00:02:06.860
JR HILDEBRAND: So
then you look at that.

00:02:06.860 --> 00:02:10.330
And I think if you
noted the radius--

00:02:10.330 --> 00:02:13.980
if you drew a full circle
out of each of those arcs--

00:02:13.980 --> 00:02:14.980
SAL KHAN: Let's do that.

00:02:14.980 --> 00:02:17.188
So let's say that this is
the shortest distance path.

00:02:17.188 --> 00:02:19.859
This is kind of a circle that
looks something like this.

00:02:19.859 --> 00:02:21.400
Let me scroll over
a little bit so we

00:02:21.400 --> 00:02:22.670
can see a little bit better.

00:02:22.670 --> 00:02:25.310
So this would be
a circle like this

00:02:25.310 --> 00:02:27.220
if you were to keep that arc.

00:02:27.220 --> 00:02:29.400
It would be a circle that
looks something like this.

00:02:29.400 --> 00:02:31.280
JR HILDEBRAND: So that's
a pretty small circle

00:02:31.280 --> 00:02:32.450
in the grand scheme
of things here, yeah.

00:02:32.450 --> 00:02:33.050
SAL KHAN: That's a small circle.

00:02:33.050 --> 00:02:34.466
And for the larger
one, the circle

00:02:34.466 --> 00:02:37.430
would look something like this.

00:02:37.430 --> 00:02:40.100
So you have a larger radius,
a larger turning radius.

00:02:40.100 --> 00:02:43.560
So you would have to have
less centripetal acceleration,

00:02:43.560 --> 00:02:45.960
inward acceleration,
and fewer g-forces

00:02:45.960 --> 00:02:48.190
on this outside one, the
larger the circle is.

00:02:48.190 --> 00:02:48.320
JR HILDEBRAND: Right.

00:02:48.320 --> 00:02:49.310
And a different
way to look at it,

00:02:49.310 --> 00:02:52.036
if you looked at the car trying
to just go around these two

00:02:52.036 --> 00:02:53.410
different circles,
and it's going

00:02:53.410 --> 00:02:55.190
to be going the same
speed on either one,

00:02:55.190 --> 00:02:59.070
it's doing a lot less work to
get around this outside circle.

00:02:59.070 --> 00:03:02.090
And therefore the speed that
you could carry around that,

00:03:02.090 --> 00:03:03.230
that sort of goes up.

00:03:03.230 --> 00:03:05.800
The car has a limited ability
to stick to the racetrack.

00:03:05.800 --> 00:03:08.040
So opening that up definitely
makes a difference.

00:03:08.040 --> 00:03:08.750
SAL KHAN: But that's
an important point.

00:03:08.750 --> 00:03:11.462
At least in Indianapolis, you're
full throttle the entire way.

00:03:11.462 --> 00:03:13.170
I mean, obviously, if
you hit the brakes,

00:03:13.170 --> 00:03:17.100
the car could do a very
small turning radius.

00:03:17.100 --> 00:03:19.130
But you're at full throttle.

00:03:19.130 --> 00:03:21.530
You're not going to have any
chance if you at all let off

00:03:21.530 --> 00:03:21.970
the gas.

00:03:21.970 --> 00:03:22.300
JR HILDEBRAND: That's right.

00:03:22.300 --> 00:03:23.674
When you qualify
at Indianapolis,

00:03:23.674 --> 00:03:26.760
you've got to put in four
laps, four of your best laps

00:03:26.760 --> 00:03:30.340
of the season, of your career
in Indianapolis to qualify.

00:03:30.340 --> 00:03:32.320
And that you are
absolutely flat trap

00:03:32.320 --> 00:03:33.150
all the way around
the racetrack.

00:03:33.150 --> 00:03:33.941
There's no lifting.

00:03:33.941 --> 00:03:34.930
There's no braking.

00:03:34.930 --> 00:03:35.480
SAL KHAN: And so
that's why you're

00:03:35.480 --> 00:03:36.700
saying the car just
wouldn't do that.

00:03:36.700 --> 00:03:38.050
If you're going all
out, the car just

00:03:38.050 --> 00:03:39.716
wouldn't even be able
to make this path.

00:03:39.716 --> 00:03:40.770
JR HILDEBRAND: Exactly.

00:03:40.770 --> 00:03:41.603
That's a good point.

00:03:41.603 --> 00:03:43.180
From the driver's
perspective, you

00:03:43.180 --> 00:03:45.347
have to stay flat out if
you're going to go fast.

00:03:45.347 --> 00:03:47.430
If you're going to set a
lap time that's relevant,

00:03:47.430 --> 00:03:49.020
you have to be able
to stay flat out.

00:03:49.020 --> 00:03:51.020
And so at that point,
you're searching

00:03:51.020 --> 00:03:52.790
for the line around
the race track

00:03:52.790 --> 00:03:55.110
that you can do that
most efficiently.

00:03:55.110 --> 00:03:58.880
And so then, in this
example, increasing

00:03:58.880 --> 00:04:01.620
that radius by going
from our green circle

00:04:01.620 --> 00:04:04.724
out to the purple circle
does that rather effectively.

00:04:04.724 --> 00:04:05.390
SAL KHAN: I see.

00:04:05.390 --> 00:04:07.223
We're going for the
purple to the green back

00:04:07.223 --> 00:04:08.967
to-- so you're saying like this.

00:04:08.967 --> 00:04:10.050
JR HILDEBRAND: Well, yeah.

00:04:10.050 --> 00:04:13.250
And so then to find the actual
optimal line, what we end up

00:04:13.250 --> 00:04:15.640
doing is starting out on
the outside of the track,

00:04:15.640 --> 00:04:19.040
then bending the car into
the inside of the track,

00:04:19.040 --> 00:04:21.190
and going back to the
outside of the track,

00:04:21.190 --> 00:04:24.834
really using all of the
road that's available to us.

00:04:24.834 --> 00:04:25.500
SAL KHAN: Right.

00:04:25.500 --> 00:04:26.416
So that's interesting.

00:04:26.416 --> 00:04:28.270
So when I posed the
question, it was

00:04:28.270 --> 00:04:31.720
kind of like my brain was just
looking at these two circles.

00:04:31.720 --> 00:04:35.060
But you realize there's a bigger
circle that you could fit here,

00:04:35.060 --> 00:04:36.800
that there's an arc like this.

00:04:36.800 --> 00:04:38.316
And this would be,
if you imagine,

00:04:38.316 --> 00:04:39.940
this would be a part
of a circle that's

00:04:39.940 --> 00:04:44.295
way huger than even that purple
circle that we're drawing.

00:04:44.295 --> 00:04:47.200
So that center of that circle
is like here or something.

00:04:47.200 --> 00:04:50.670
So you have a lot less
centripetal acceleration

00:04:50.670 --> 00:04:52.530
that you have to place,
inward acceleration

00:04:52.530 --> 00:04:53.550
that you have to
place on the car.

00:04:53.550 --> 00:04:54.508
JR HILDEBRAND: Exactly.

00:04:54.508 --> 00:04:56.840
And therefore, the
car is able to carry

00:04:56.840 --> 00:05:00.012
a massively increased level
of speed through the corner.

00:05:00.012 --> 00:05:01.720
And that's really what
we're looking for.

00:05:01.720 --> 00:05:05.220
And you consider, I think
it's a very interesting-- when

00:05:05.220 --> 00:05:07.180
I think about what I'm
doing as the driver,

00:05:07.180 --> 00:05:10.200
I don't think I
really am consciously

00:05:10.200 --> 00:05:13.000
thinking that much about
the mathematics that

00:05:13.000 --> 00:05:15.620
go into finding this
optimal racing line.

00:05:15.620 --> 00:05:18.100
You sort of instinctually
just gravitate

00:05:18.100 --> 00:05:22.096
towards what the car
feels like it wants to do.

00:05:22.096 --> 00:05:23.970
But when we look at it
from this perspective,

00:05:23.970 --> 00:05:27.010
you've got the car going
down the straight away here.

00:05:27.010 --> 00:05:29.000
It's at 240 miles per hour.

00:05:29.000 --> 00:05:31.160
That's almost as fast as
the car is going to go.

00:05:31.160 --> 00:05:33.780
So it's just this sort
of terminal velocity.

00:05:33.780 --> 00:05:36.149
The drag of the
air hitting the car

00:05:36.149 --> 00:05:37.940
won't allow it to go
much faster than that.

00:05:37.940 --> 00:05:39.750
SAL KHAN: The engine's
giving all the power it can.

00:05:39.750 --> 00:05:39.960
JR HILDEBRAND: Yeah.

00:05:39.960 --> 00:05:40.640
You're absolutely flat out.

00:05:40.640 --> 00:05:42.270
SAL KHAN: And that's just
offsetting the drag of that,

00:05:42.270 --> 00:05:44.420
so that you can't accelerate
to that top speed.

00:05:44.420 --> 00:05:44.720
JR HILDEBRAND: Exactly.

00:05:44.720 --> 00:05:45.610
It's almost like you're hitting
a wall of air at that point.

00:05:45.610 --> 00:05:48.620
You're not going to be able
to accelerate any faster.

00:05:48.620 --> 00:05:50.230
And so what you're
really trying to do

00:05:50.230 --> 00:05:53.110
is you're trying to-- in
order to set that fastest lap

00:05:53.110 --> 00:05:56.930
time, which ends up
equating to the highest

00:05:56.930 --> 00:05:59.680
average speed around the
lap, that's what's the lowest

00:05:59.680 --> 00:06:02.060
number in terms of lap time
perspective-- you're trying

00:06:02.060 --> 00:06:05.230
to get the car to
most efficiently get

00:06:05.230 --> 00:06:08.220
through the corners so that you
can allow it to accelerate down

00:06:08.220 --> 00:06:10.090
the straights as
much as you can.

00:06:10.090 --> 00:06:13.750
You're getting it to diverge
from this intended course that

00:06:13.750 --> 00:06:16.280
is going on here as
efficiently as you can.

00:06:16.280 --> 00:06:19.860
And so by creating the largest
radius around the corner,

00:06:19.860 --> 00:06:22.170
that's how we end up
finding that optimal line.

00:06:22.170 --> 00:06:23.860
SAL KHAN: That's fascinating.


Structure in rational expression

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.290
Let's say we we're told that
the position of a particle

00:00:03.290 --> 00:00:06.630
can be figured out
from, or the position

00:00:06.630 --> 00:00:08.370
of a particle as
a function of time

00:00:08.370 --> 00:00:09.959
is given by this
crazy expression.

00:00:09.959 --> 00:00:11.750
The position could be
positive or negative.

00:00:11.750 --> 00:00:14.400
And this expression is t minus
a, times a minus t, times

00:00:14.400 --> 00:00:17.700
t minus a, times t
minus b, all of that

00:00:17.700 --> 00:00:20.580
over the square root of
a squared plus b squared.

00:00:20.580 --> 00:00:22.230
And they also tell
us that c is greater

00:00:22.230 --> 00:00:26.540
than b, is greater than
a, is greater than 0.

00:00:26.540 --> 00:00:29.900
So given that information,
we have two statements

00:00:29.900 --> 00:00:30.850
right over here.

00:00:30.850 --> 00:00:34.820
We have the position at times
c on the left hand side.

00:00:34.820 --> 00:00:37.330
And over here, we have
the number of times

00:00:37.330 --> 00:00:42.550
that our function is equal to 0,
the number of times that p of t

00:00:42.550 --> 00:00:43.900
is equal to 0.

00:00:43.900 --> 00:00:47.270
What I want you to do is
pause the video right now

00:00:47.270 --> 00:00:49.800
and think about which
of these statements

00:00:49.800 --> 00:00:51.420
provides a larger value.

00:00:51.420 --> 00:00:55.970
Is p of c greater than or less
than the number of times p of t

00:00:55.970 --> 00:00:57.010
equals 0?

00:00:57.010 --> 00:01:00.640
And I guess a third and fourth
option would be that you maybe

00:01:00.640 --> 00:01:02.640
don't have enough information
to figure this out

00:01:02.640 --> 00:01:04.140
or maybe that these are equal.

00:01:04.140 --> 00:01:06.490
So I encourage you to
figure that out right now.

00:01:06.490 --> 00:01:07.630
Which of these are larger?

00:01:07.630 --> 00:01:09.213
Or do you not have
enough information?

00:01:09.213 --> 00:01:11.170
Or are they equal?

00:01:11.170 --> 00:01:13.630
So I'm assuming you've
given a go at it.

00:01:13.630 --> 00:01:15.630
So let's think
about each of them.

00:01:15.630 --> 00:01:18.530
So let's think about
what p of c is.

00:01:18.530 --> 00:01:21.520
So p of c is going to
be equal to, right here

00:01:21.520 --> 00:01:24.480
p-- I don't want to
arbitrarily switch

00:01:24.480 --> 00:01:26.260
colors, which I sometimes do.

00:01:26.260 --> 00:01:32.680
p of c is going to be
equal to, let's see.

00:01:32.680 --> 00:01:34.440
It's going to be c minus a.

00:01:34.440 --> 00:01:36.810
I'll do this all
in this one color.

00:01:36.810 --> 00:01:42.880
It's going to be c minus
a, times a minus c, times

00:01:42.880 --> 00:01:48.700
c minus a, times c
minus b, all of that

00:01:48.700 --> 00:01:54.860
over the square root of
a squared plus b squared.

00:01:54.860 --> 00:01:56.774
So what do we know
about this quantity?

00:01:56.774 --> 00:01:58.940
What do we know about this
quantity right over here?

00:01:58.940 --> 00:02:00.315
Let me highlight
all the Cs here.

00:02:00.315 --> 00:02:05.860
So c minus a, a minus
c, c minus a, c minus b.

00:02:05.860 --> 00:02:09.199
Well, they tell us that
c is larger than a and b

00:02:09.199 --> 00:02:11.209
and that they're all positive.

00:02:11.209 --> 00:02:13.000
So maybe we can come
up with some statement

00:02:13.000 --> 00:02:15.550
about whether this thing
is positive or negative,

00:02:15.550 --> 00:02:17.650
whether this expression is.

00:02:17.650 --> 00:02:19.930
So what's c minus a going to be?

00:02:19.930 --> 00:02:22.300
Well, c is greater than a.

00:02:22.300 --> 00:02:24.790
So this is going to be positive.

00:02:24.790 --> 00:02:27.280
What about a minus c?

00:02:27.280 --> 00:02:29.100
Well, a is less than c.

00:02:29.100 --> 00:02:32.010
So this is going to result
in a negative number.

00:02:32.010 --> 00:02:35.130
c minus a, well, this is
going to be a positive again.

00:02:35.130 --> 00:02:38.170
And then c minus b is also
going to be a positive.

00:02:38.170 --> 00:02:41.487
c is greater than both b and a.

00:02:41.487 --> 00:02:43.320
And what do we have
here in the denominator?

00:02:43.320 --> 00:02:45.028
Well, the square root
of a squared plus b

00:02:45.028 --> 00:02:49.370
squared, well, this is just
going to be a positive value.

00:02:49.370 --> 00:02:50.850
So what do we have
going on here?

00:02:50.850 --> 00:02:53.040
Here in the numerator, I have
a positive times a negative,

00:02:53.040 --> 00:02:54.580
times a positive,
times a positive.

00:02:54.580 --> 00:02:56.200
So what's that going to be?

00:02:56.200 --> 00:02:58.242
Well, that's going
to be a negative.

00:02:58.242 --> 00:03:00.700
A positive, times a positive,
times a positive is positive.

00:03:00.700 --> 00:03:02.450
And then you throw
that negative in there.

00:03:02.450 --> 00:03:07.840
So you're going to get a
negative over a positive.

00:03:07.840 --> 00:03:10.290
And what's a negative
divided by a positive?

00:03:10.290 --> 00:03:12.560
Well, that's going
to be a negative.

00:03:15.370 --> 00:03:17.470
So we don't know what
the actual value is.

00:03:17.470 --> 00:03:19.790
But all we do know is
that this provides us

00:03:19.790 --> 00:03:23.240
with a negative value.

00:03:23.240 --> 00:03:24.872
So this ends up being
a positive value.

00:03:24.872 --> 00:03:27.330
Then we could make a statement
that this, we just say, hey,

00:03:27.330 --> 00:03:28.770
this is a negative
value as well.

00:03:28.770 --> 00:03:30.478
But we might not have
enough information.

00:03:30.478 --> 00:03:34.190
So let's think about the number
of times p of t is equal to 0.

00:03:34.190 --> 00:03:36.600
Well, p of t is
equal to 0 whenever

00:03:36.600 --> 00:03:38.840
the numerator right
over here is equal to 0.

00:03:38.840 --> 00:03:41.190
And when would the
numerator equal 0?

00:03:41.190 --> 00:03:47.940
Well, I have the product of
this 1, 2, 3, 4 expression.

00:03:47.940 --> 00:03:50.640
So if any one of these
expressions is 0,

00:03:50.640 --> 00:03:52.570
then the entire numerator
is going to be 0.

00:03:52.570 --> 00:03:55.740
So let's think about how you
can make these expressions 0.

00:03:55.740 --> 00:03:57.100
So the expression could be 0.

00:03:57.100 --> 00:04:00.340
This will be 0 if
t is equal to a.

00:04:00.340 --> 00:04:02.880
This would also be 0
if t is equal to a.

00:04:02.880 --> 00:04:05.440
And this would also be
0 if t is equal to a.

00:04:05.440 --> 00:04:09.450
And this would be 0
if t is equal to b.

00:04:09.450 --> 00:04:12.860
So there's two values
for t that will

00:04:12.860 --> 00:04:18.180
make this numerator equal to
0, t equals a or t equals b.

00:04:18.180 --> 00:04:19.589
So there's two times.

00:04:19.589 --> 00:04:25.345
So the number of times that
p of t is equal to 0 is two.

00:04:25.345 --> 00:04:26.720
So now let's answer
our question.

00:04:26.720 --> 00:04:30.110
What is larger, the number
2, is the number positive 2,

00:04:30.110 --> 00:04:33.230
versus some negative number?

00:04:33.230 --> 00:04:36.000
Well, 2 is larger than
any negative number.

00:04:36.000 --> 00:04:40.630
So this is the larger quantity.


Structure in exponential expression

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.270
Let's say that the
position of some particle

00:00:03.270 --> 00:00:06.794
as a function of time is
given by this expression

00:00:06.794 --> 00:00:07.460
right over here.

00:00:07.460 --> 00:00:10.220
Negative d to the
negative t power

00:00:10.220 --> 00:00:13.230
plus c to the fourth
over c squared

00:00:13.230 --> 00:00:17.320
plus 1, where c and d are
constants and both of them

00:00:17.320 --> 00:00:18.577
are greater than one.

00:00:18.577 --> 00:00:20.660
So what I want to do over
the course of this video

00:00:20.660 --> 00:00:24.420
is see what can we infer
based on this expression,

00:00:24.420 --> 00:00:26.567
this function definition,
that we have here.

00:00:26.567 --> 00:00:28.650
And the first thing that
I want you to think about

00:00:28.650 --> 00:00:30.970
is, what is the
initial position?

00:00:30.970 --> 00:00:33.600
If I were to express the
initial position, in terms

00:00:33.600 --> 00:00:37.500
of c's and d's, and
try to simplify it.

00:00:37.500 --> 00:00:39.220
So I encourage you
to pause the video

00:00:39.220 --> 00:00:43.630
and try to find an expression
for the initial position.

00:00:43.630 --> 00:00:45.850
Well, the initial
position is the position

00:00:45.850 --> 00:00:48.930
we're at when our
time is equal to 0.

00:00:48.930 --> 00:00:54.390
So we essentially just
want to find p of 0.

00:00:54.390 --> 00:01:00.850
And p of 0 is going to
be equal to negative d

00:01:00.850 --> 00:01:02.932
to the negative 0.

00:01:02.932 --> 00:01:04.140
So I'll just write that down.

00:01:04.140 --> 00:01:14.370
Negative 0 plus c to the
fourth over c squared plus 1.

00:01:14.370 --> 00:01:16.060
Well, d to the
negative 0, that's

00:01:16.060 --> 00:01:18.510
the same thing as d
to the 0, and since we

00:01:18.510 --> 00:01:21.490
know that d is non-zero,
we know this is defined.

00:01:21.490 --> 00:01:25.190
Anything non-zero to the
0-th power is going to be 1.

00:01:25.190 --> 00:01:27.300
And the zero is
actually under debate,

00:01:27.300 --> 00:01:29.160
what 0 to the 0-th power is.

00:01:29.160 --> 00:01:31.660
But we can safely say
that this right over here

00:01:31.660 --> 00:01:33.760
is going to be equal to 1.

00:01:33.760 --> 00:01:36.940
And so the numerator
here simplifies 2.

00:01:36.940 --> 00:01:39.400
This is equal to-- and
I'll switch the order.

00:01:39.400 --> 00:01:44.930
c to the fourth minus 1
over c squared plus 1.

00:01:44.930 --> 00:01:47.730
And now this might jump out at
you as a difference of squares.

00:01:47.730 --> 00:01:51.240
We could write this
as c squared, squared,

00:01:51.240 --> 00:01:58.370
minus 1 squared over
c squared plus 1.

00:01:58.370 --> 00:02:00.470
And that's the same
thing as c squared

00:02:00.470 --> 00:02:04.870
plus 1 times c squared minus 1.

00:02:04.870 --> 00:02:08.169
All of that over
c squared plus 1.

00:02:08.169 --> 00:02:10.280
And we have a c squared
plus 1 in the numerator

00:02:10.280 --> 00:02:13.160
and that denominator
so we can simplify.

00:02:13.160 --> 00:02:17.720
And so our initial position is
going to be c squared minus 1.

00:02:17.720 --> 00:02:21.790
So that actually simplified
out quite nicely.

00:02:21.790 --> 00:02:24.397
Now the next question I'm
going to ask you is-- OK,

00:02:24.397 --> 00:02:26.730
we know that the initial
position that time equals zero,

00:02:26.730 --> 00:02:30.240
the particle is going to
be at c squared minus 1.

00:02:30.240 --> 00:02:32.310
But what happens after that?

00:02:32.310 --> 00:02:34.410
Does the position
keep increasing?

00:02:34.410 --> 00:02:36.170
Does the position
keep decreasing?

00:02:36.170 --> 00:02:38.820
Or does the position maybe
increase and then decrease,

00:02:38.820 --> 00:02:43.010
or decrease and then increase
as in keep swapping around?

00:02:43.010 --> 00:02:45.160
So I encourage you to pause
the video now and think

00:02:45.160 --> 00:02:46.670
about what happens
to the position.

00:02:46.670 --> 00:02:47.460
Does it keep increasing?

00:02:47.460 --> 00:02:48.460
Does it keep decreasing?

00:02:48.460 --> 00:02:51.630
Or does it do something else?

00:02:51.630 --> 00:02:53.710
Well, let's answer
that question of what's

00:02:53.710 --> 00:02:56.350
happening to the position
after our initial position.

00:02:56.350 --> 00:03:00.590
We really just have to focus
on this term right over here.

00:03:00.590 --> 00:03:02.460
This d to the negative t.

00:03:02.460 --> 00:03:05.010
This is the only part
that really t is driving.

00:03:05.010 --> 00:03:07.050
All these other things
are staying the same

00:03:07.050 --> 00:03:08.850
as we go through time.

00:03:08.850 --> 00:03:11.460
So what happens to d to
the negative t power,

00:03:11.460 --> 00:03:16.570
to this part of this first
term, as t goes from 0 onwards.

00:03:16.570 --> 00:03:19.180
And to think about
that, let's plot.

00:03:19.180 --> 00:03:23.090
Let's plot what the function d
to the negative t looks like.

00:03:23.090 --> 00:03:28.180
d to the negative
t would look like--

00:03:28.180 --> 00:03:29.910
and we know that d
is greater than 1.

00:03:32.640 --> 00:03:36.390
So when t is equal to 0-- So
this is t right over here.

00:03:36.390 --> 00:03:38.500
And over here we're going
to plot on this axis.

00:03:38.500 --> 00:03:42.370
We're going to plot
d to the negative t.

00:03:42.370 --> 00:03:46.170
When t is equal to 0 this
is going to be equal to 1.

00:03:46.170 --> 00:03:47.950
We've already seen that.

00:03:47.950 --> 00:03:50.470
Now what happens as t increases?

00:03:50.470 --> 00:03:52.600
Say t increases to 1.

00:03:55.547 --> 00:03:57.880
Well, now this is going to
be d to the negative 1, which

00:03:57.880 --> 00:03:59.304
is the same thing as 1 over d.

00:03:59.304 --> 00:04:00.970
And we don't know the
exact value for d,

00:04:00.970 --> 00:04:03.090
but we know, since d is
greater than 1, 1 over

00:04:03.090 --> 00:04:04.760
d is going to be less than 1.

00:04:04.760 --> 00:04:07.300
So let's say this is 1
over d right over here.

00:04:07.300 --> 00:04:08.160
1 over d.

00:04:08.160 --> 00:04:11.010
So it's going to be
something like that.

00:04:11.010 --> 00:04:13.460
And then when t is
at 2 we're going

00:04:13.460 --> 00:04:16.290
to be at 1 over d
squared, which is going

00:04:16.290 --> 00:04:19.820
to be something like
right over there.

00:04:19.820 --> 00:04:23.170
And you see at least this term
what it's doing as t increases.

00:04:23.170 --> 00:04:25.890
As t increases, d
to the negative t

00:04:25.890 --> 00:04:27.135
is strictly decreasing.

00:04:31.580 --> 00:04:35.240
And once again we know that
because d is greater than 1.

00:04:35.240 --> 00:04:38.980
So this term right over
here is strictly decreasing.

00:04:38.980 --> 00:04:41.570
So this is decreasing.

00:04:41.570 --> 00:04:42.790
That part of it.

00:04:42.790 --> 00:04:43.800
But we're not adding it.

00:04:43.800 --> 00:04:45.480
We're subtracting it.

00:04:45.480 --> 00:04:49.020
We're subtracting
from the beginning.

00:04:49.020 --> 00:04:50.620
At first this starts off at 1.

00:04:50.620 --> 00:04:51.810
We subtract 1.

00:04:51.810 --> 00:04:54.720
And then we start subtracting
smaller and smaller things

00:04:54.720 --> 00:04:56.130
than 1.

00:04:56.130 --> 00:05:00.030
So if this is decreasing,
but we're subtracting it,

00:05:00.030 --> 00:05:02.840
we're subtracting smaller
and smaller things,

00:05:02.840 --> 00:05:05.250
this whole thing,
the negative of it,

00:05:05.250 --> 00:05:06.725
is going to be increasing.

00:05:11.331 --> 00:05:12.580
Another way to think about it.

00:05:12.580 --> 00:05:16.200
If you wanted to plot
negative d to the negative t,

00:05:16.200 --> 00:05:18.300
it would look like this.

00:05:18.300 --> 00:05:20.880
It would just be the negative
of this right over here.

00:05:20.880 --> 00:05:24.693
So it would look
something like-- I'll

00:05:24.693 --> 00:05:28.980
do that yellow color-- this.

00:05:31.880 --> 00:05:33.580
So this whole term
right over here,

00:05:33.580 --> 00:05:39.530
negative d to the negative
t, is constantly increasing.

00:05:39.530 --> 00:05:42.050
And we know that all of these
other things, well, these

00:05:42.050 --> 00:05:43.450
are just going to be fixed.

00:05:43.450 --> 00:05:46.580
So this entire
expression is constantly

00:05:46.580 --> 00:05:48.630
increasing starting
at t equals 0.

00:05:48.630 --> 00:05:53.190
And then t going into larger
and larger positive values.

00:05:53.190 --> 00:05:54.710
Now the last thing
I want to ask you

00:05:54.710 --> 00:05:57.080
is, what is the
maximum value here?

00:05:57.080 --> 00:05:59.722
What is a value that this
will never be able to get to?

00:05:59.722 --> 00:06:01.680
It might try to get close
to it, but it's never

00:06:01.680 --> 00:06:03.350
going to quite get there.

00:06:03.350 --> 00:06:05.300
Well, we already know
that it's increasing,

00:06:05.300 --> 00:06:09.342
but let's just think about what
happens as t becomes really,

00:06:09.342 --> 00:06:10.550
really, really large numbers.

00:06:10.550 --> 00:06:13.700
Really you could think about
it as t approaches infinity.

00:06:13.700 --> 00:06:16.640
Well, once again let's look at
this d to the negative t term.

00:06:16.640 --> 00:06:20.940
You see the d to the negative
t, as t gets larger and larger,

00:06:20.940 --> 00:06:22.770
is getting smaller and smaller.

00:06:22.770 --> 00:06:29.450
This term right over
here is approaching 0

00:06:29.450 --> 00:06:34.920
as t goes to infinity.

00:06:34.920 --> 00:06:38.440
Well, if this is approaching 0
that means we're subtracting 0.

00:06:38.440 --> 00:06:41.886
So this whole yellow thing, the
negative d to the negative t,

00:06:41.886 --> 00:06:43.260
is increasing but
it's increasing

00:06:43.260 --> 00:06:44.937
at a smaller and slower rate.

00:06:44.937 --> 00:06:46.770
So this is the negative
d to the negative t.

00:06:46.770 --> 00:06:47.769
This is right over here.

00:06:47.769 --> 00:06:49.410
You see it increases
but it never quite

00:06:49.410 --> 00:06:54.430
gets to the horizontal
axis right over here.

00:06:54.430 --> 00:06:57.970
And so if we think about it
as t approaches infinity,

00:06:57.970 --> 00:07:01.090
this whole thing just becomes
0 and our entire position

00:07:01.090 --> 00:07:03.230
is approaching but
never quite gets

00:07:03.230 --> 00:07:09.446
to c to the fourth
over c squared plus 1.

00:07:09.446 --> 00:07:11.570
So one way to think about
it, it's approaching this

00:07:11.570 --> 00:07:15.840
but it never quite gets to a
position of c to the fourth

00:07:15.840 --> 00:07:18.800
over c squared plus 1.


Formula for a finite geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:02.640
What I hope to do in
this video is derive

00:00:02.640 --> 00:00:06.850
a formula for the sum of
a finite geometric series.

00:00:06.850 --> 00:00:09.720
And right here is a
finite geometric series.

00:00:09.720 --> 00:00:11.600
I start with some first term a.

00:00:11.600 --> 00:00:14.820
Notice, I wrote it as
a times r to the 0.

00:00:14.820 --> 00:00:17.100
But r to the 0-- for any r.

00:00:17.100 --> 00:00:18.980
We're assuming r is non-zero.

00:00:18.980 --> 00:00:21.170
This is going to be
1 right over here.

00:00:21.170 --> 00:00:23.240
So our first term
is going to be a.

00:00:23.240 --> 00:00:25.660
And then each successive
term is a previous term

00:00:25.660 --> 00:00:28.060
times our common ratio, times r.

00:00:28.060 --> 00:00:30.810
So a times r is a
times r to the first.

00:00:30.810 --> 00:00:32.820
That times r is ar squared.

00:00:32.820 --> 00:00:35.540
On and on, all the way until
we go a times r to the n.

00:00:35.540 --> 00:00:38.950
This is a finite
series right over here.

00:00:38.950 --> 00:00:42.960
So let's just
define this series.

00:00:42.960 --> 00:00:47.050
I'll use the letter
S sub n as being

00:00:47.050 --> 00:00:50.480
equal to all of this
business right over here.

00:00:50.480 --> 00:00:55.020
Now, let me multiply S sub
n times our common ratio.

00:00:55.020 --> 00:00:56.710
So let me write this
right over here.

00:00:56.710 --> 00:00:59.707
So let's say we
have r-- I'm going

00:00:59.707 --> 00:01:07.810
to do the same color as our
previous r-- r times S sub n.

00:01:07.810 --> 00:01:10.290
And what is that
going to be equal to?

00:01:10.290 --> 00:01:13.580
Well, we're just going to
multiply each term here by r.

00:01:13.580 --> 00:01:17.790
And we could even multiply this
thing by r if we want as well.

00:01:17.790 --> 00:01:28.939
This would now be the sum from
k equals 0 to n of a times r.

00:01:28.939 --> 00:01:30.480
But now, instead of
r to the k, we're

00:01:30.480 --> 00:01:32.940
now multiplying each of
these terms times r again.

00:01:32.940 --> 00:01:36.990
So it's going to be
the k plus 1 power.

00:01:36.990 --> 00:01:39.800
Or-- and it's a little easier to
conceptualize right over here--

00:01:39.800 --> 00:01:43.030
let's just multiply each
of these terms times r.

00:01:43.030 --> 00:01:48.130
So a times r to the
0 times r is going

00:01:48.130 --> 00:01:50.440
to be ar to the first power.

00:01:50.440 --> 00:01:55.746
So that's going to be a
times r to the first power.

00:01:55.746 --> 00:01:58.185
Let me write that down. a
times r to the first power.

00:01:58.185 --> 00:02:00.650
Or just a times r because,
obviously, r to the first

00:02:00.650 --> 00:02:01.890
is just r.

00:02:01.890 --> 00:02:06.290
And then, plus-- let's
multiply this term times r.

00:02:06.290 --> 00:02:08.514
So plus a times r squared.

00:02:13.430 --> 00:02:15.130
And we'll just keep on going.

00:02:19.190 --> 00:02:23.100
We're going to get all the way
to a times r to the n plus 1.

00:02:23.100 --> 00:02:24.460
So let me write it like this.

00:02:24.460 --> 00:02:27.510
So we're going to go to
plus-- and typically, you'd

00:02:27.510 --> 00:02:28.646
only write three dots.

00:02:28.646 --> 00:02:29.895
I went overboard with my dots.

00:02:29.895 --> 00:02:32.126
So let me do it-- dot, dot, dot.

00:02:32.126 --> 00:02:44.430
So plus a times r to the n
plus a times r to the n plus 1.

00:02:49.910 --> 00:02:51.660
So notice, I just took
each of these terms

00:02:51.660 --> 00:02:52.955
and multiplied them by r.

00:02:52.955 --> 00:02:54.260
You see that right over here.

00:02:54.260 --> 00:02:55.690
Multiply by r, get that term.

00:02:55.690 --> 00:02:57.500
Multiply by r, get that term.

00:02:57.500 --> 00:02:59.750
And then, I don't show the
other ones multiplied by r,

00:02:59.750 --> 00:03:02.290
but all the way to
multiply this one by r.

00:03:02.290 --> 00:03:05.130
And you get this
term right over here.

00:03:05.130 --> 00:03:07.130
Now, what I want to do
is something really neat.

00:03:07.130 --> 00:03:10.460
And this is one of my
favorite derivations

00:03:10.460 --> 00:03:11.960
in all of mathematics
because you're

00:03:11.960 --> 00:03:14.430
able to do something
pretty neat at this stage.

00:03:14.430 --> 00:03:18.680
What happens if I were to
subtract r times S sub n

00:03:18.680 --> 00:03:20.087
from S sub n?

00:03:20.087 --> 00:03:21.170
So let's think about that.

00:03:21.170 --> 00:03:27.210
Well, over here, we would just
get S sub n minus r times S sub

00:03:27.210 --> 00:03:27.710
n.

00:03:31.237 --> 00:03:33.320
So notice, all I'm doing
is I'm starting with this

00:03:33.320 --> 00:03:35.807
and I'm subtracting
the same thing

00:03:35.807 --> 00:03:37.140
from both sides of the equation.

00:03:37.140 --> 00:03:40.680
On the left-hand side, I'm
subtracting r times S sub n.

00:03:40.680 --> 00:03:42.770
And on the right-hand
side, from this

00:03:42.770 --> 00:03:44.810
I'm going to
subtract all of this.

00:03:44.810 --> 00:03:46.680
So what do we get on
the right-hand side?

00:03:46.680 --> 00:03:47.710
So this is going
to be equal to--

00:03:47.710 --> 00:03:49.209
and I'm going to
ignore this for now

00:03:49.209 --> 00:03:51.610
because what we really care
about is this business.

00:03:51.610 --> 00:03:53.734
So a times r.

00:03:53.734 --> 00:03:55.900
What I'm going to do is I'm
just going to cancel out

00:03:55.900 --> 00:03:58.200
all of the terms that
have the same degree.

00:03:58.200 --> 00:04:01.870
So for example, a
times r to the 0.

00:04:01.870 --> 00:04:04.780
There's no other r
to the 0 term here.

00:04:04.780 --> 00:04:07.400
So we're just going to have
that a times r to the 0.

00:04:07.400 --> 00:04:10.080
Or I could just write that
as a, obviously. r to the 0

00:04:10.080 --> 00:04:11.360
is going to be 1.

00:04:11.360 --> 00:04:12.982
So I'll just write that as a.

00:04:12.982 --> 00:04:15.170
That's that right over there.

00:04:15.170 --> 00:04:17.380
And then, we have a times
r to the first power

00:04:17.380 --> 00:04:19.370
minus r to the first power.

00:04:19.370 --> 00:04:21.470
Well, that's just
going to cancel out.

00:04:21.470 --> 00:04:22.990
These are going to cancel out.

00:04:22.990 --> 00:04:24.490
They're going to
keep canceling out.

00:04:24.490 --> 00:04:26.820
They're going to keep
canceling out all the way

00:04:26.820 --> 00:04:28.570
until all we have
left over is this.

00:04:28.570 --> 00:04:30.540
And there's nothing
to subtract this from.

00:04:30.540 --> 00:04:33.890
We have no a times r to
the n plus 1 power up here.

00:04:33.890 --> 00:04:35.760
So we're just going
to subtract it.

00:04:35.760 --> 00:04:45.510
So minus a times r to
the n plus 1 power.

00:04:49.217 --> 00:04:50.800
And now we can just
solve for our sum.

00:04:50.800 --> 00:04:52.716
Remember, that's the
whole point of this video

00:04:52.716 --> 00:04:56.640
to begin with-- to try to
derive a formula for this sum.

00:04:56.640 --> 00:05:00.850
So let's factor
out S sub n here.

00:05:00.850 --> 00:05:08.770
So this is S sub
n times 1 minus r

00:05:08.770 --> 00:05:11.530
is equal to this
right over here.

00:05:11.530 --> 00:05:17.560
It's equal to the first term
minus the first term times r

00:05:17.560 --> 00:05:18.880
to the n plus 1.

00:05:18.880 --> 00:05:20.610
So one more power
times the number

00:05:20.610 --> 00:05:23.470
of terms in our original sum.

00:05:23.470 --> 00:05:29.580
So let's do that-- times
r to the n plus 1 power.

00:05:29.580 --> 00:05:31.730
And so to solve for
S sub n, we just

00:05:31.730 --> 00:05:33.990
have to divide both
sides by 1 minus r.

00:05:39.740 --> 00:05:43.610
And so we are left with-- if
we assume r is not equal to 1

00:05:43.610 --> 00:05:57.670
in this case-- we are left with
S sub n is equal to a minus

00:05:57.670 --> 00:06:02.330
a times r-- I'm just rewriting
what I have up here-- r

00:06:02.330 --> 00:06:09.994
to the n plus 1 over 1 minus r.

00:06:09.994 --> 00:06:10.910
And you say, hey, Sal.

00:06:10.910 --> 00:06:13.060
Well, why is this
at all interesting?

00:06:13.060 --> 00:06:14.480
Why is this useful?

00:06:14.480 --> 00:06:18.780
Well, let's actually
apply this formula

00:06:18.780 --> 00:06:22.210
that we just finished deriving.

00:06:22.210 --> 00:06:25.950
Let's say that we have
a geometric series.

00:06:25.950 --> 00:06:29.180
So let's say our first
term is-- oh, I don't know.

00:06:29.180 --> 00:06:32.990
Let's say our first term is 3.

00:06:32.990 --> 00:06:35.510
I'll do the same colors
as I was doing before.

00:06:35.510 --> 00:06:44.760
It is 3 times 1/2
to the k-th power.

00:06:44.760 --> 00:06:48.420
And we're going to start at
k equals 0 all the way to-- I

00:06:48.420 --> 00:06:51.360
don't know-- k equals 100.

00:06:51.360 --> 00:06:55.410
And just to visualize
what this looks like,

00:06:55.410 --> 00:06:57.630
the first term is just
going to be 3 times

00:06:57.630 --> 00:06:59.000
one half to the 0 power.

00:06:59.000 --> 00:06:59.610
It's just 3.

00:06:59.610 --> 00:07:02.670
So our first term is
3, as was intended.

00:07:02.670 --> 00:07:04.430
So our first term is 3.

00:07:04.430 --> 00:07:08.400
Plus 3 times 1/2 to
the first power now.

00:07:08.400 --> 00:07:10.890
This is 3 times 1/2
to the 0-th power.

00:07:10.890 --> 00:07:13.650
Now, 3 times 1/2
to the first power.

00:07:13.650 --> 00:07:15.520
3 times 1/2.

00:07:15.520 --> 00:07:17.670
I could write a first
power there if I'd like.

00:07:17.670 --> 00:07:21.400
And then, plus 3 times
1/2 half squared.

00:07:24.940 --> 00:07:30.200
And we'll just keep on going
all the way-- I'll do it

00:07:30.200 --> 00:07:32.170
right over here--
plus, we're going

00:07:32.170 --> 00:07:36.400
to keep on going all the way
until we get to 3 times 1/2

00:07:36.400 --> 00:07:37.420
to the 100th power.

00:07:40.662 --> 00:07:42.370
Now, you could imagine
you could probably

00:07:42.370 --> 00:07:44.310
calculate this
with a calculator,

00:07:44.310 --> 00:07:46.260
but it would get
very, very hard.

00:07:46.260 --> 00:07:48.250
And it would be very,
very time-consuming

00:07:48.250 --> 00:07:52.600
to add up these 101
terms right over here.

00:07:52.600 --> 00:07:54.860
Instead, we can
apply this formula.

00:07:54.860 --> 00:07:57.690
So this sum right
over here is going

00:07:57.690 --> 00:08:00.990
to be equal to-- if
we were to add it up--

00:08:00.990 --> 00:08:03.470
it's going to be equal to
our first term, which is 3.

00:08:03.470 --> 00:08:11.020
It's going to be equal to
3 minus 3 times 1/2-- 1/2

00:08:11.020 --> 00:08:14.590
is our common ratio here--
times 1/2 to the n plus 1.

00:08:14.590 --> 00:08:15.570
Well, n is 100.

00:08:15.570 --> 00:08:17.105
So it's to the 101 power.

00:08:19.980 --> 00:08:26.740
All of that over 1 minus 1/2.

00:08:26.740 --> 00:08:29.150
And now this is much easier.

00:08:29.150 --> 00:08:31.330
It would be hard to
do it in your head,

00:08:31.330 --> 00:08:34.340
but this is much easier
to type into a calculator

00:08:34.340 --> 00:08:36.450
and actually get a
reasonable answer.

00:08:36.450 --> 00:08:37.940
So let's actually do that.

00:08:37.940 --> 00:08:39.890
It'll feel satisfying.

00:08:39.890 --> 00:08:41.070
So let's clear this.

00:08:41.070 --> 00:08:46.660
So it's going to be 3 minus
3 times-- I could write 1/2

00:08:46.660 --> 00:08:52.420
as 0.5-- 0.5 to the 101 power.

00:08:55.925 --> 00:08:57.300
Did I get that
[INAUDIBLE] right?

00:08:57.300 --> 00:09:03.640
3 minus 3 times 0.5 to the
101 power divided by-- well,

00:09:03.640 --> 00:09:06.150
1 minus 1/2 is just
1/2-- so it's just

00:09:06.150 --> 00:09:10.240
going to be divided by
1/2 or divided by 0.5.

00:09:10.240 --> 00:09:12.960
And we get 6.

00:09:12.960 --> 00:09:14.100
We get 6.

00:09:14.100 --> 00:09:16.515
And it's not going
to be exactly 6.

00:09:16.515 --> 00:09:17.890
And we're going
to see that, when

00:09:17.890 --> 00:09:21.770
we take the infinite sum,
that's going to be exactly 6.

00:09:21.770 --> 00:09:23.400
But it's going to be
pretty darn close.

00:09:23.400 --> 00:09:26.830
We're actually, we're hitting
up against the precision

00:09:26.830 --> 00:09:27.637
of the calculator.

00:09:27.637 --> 00:09:29.470
So based on the precision
of the calculator,

00:09:29.470 --> 00:09:34.840
we get, it's going to
be pretty close to 6.


Sigma notation for sums

WEBVTT
Kind: captions
Language: en

00:00:00.251 --> 00:00:02.250
What I want to do in this
video is introduce you

00:00:02.250 --> 00:00:05.500
to the idea of Sigma notation,
which will be used extensively

00:00:05.500 --> 00:00:07.780
through your
mathematical career.

00:00:07.780 --> 00:00:11.700
So let's just say you wanted
to find a sum of some terms,

00:00:11.700 --> 00:00:13.430
and these terms have a pattern.

00:00:13.430 --> 00:00:15.830
So let's say you want to
find the sum of the first 10

00:00:15.830 --> 00:00:16.329
numbers.

00:00:16.329 --> 00:00:20.320
So you could say 1
plus 2 plus 3 plus,

00:00:20.320 --> 00:00:24.280
and you go all the
way to plus 9 plus 10.

00:00:24.280 --> 00:00:26.776
And I clearly could have even
written this whole thing out,

00:00:26.776 --> 00:00:29.150
but you can imagine it becomes
a lot harder if you wanted

00:00:29.150 --> 00:00:31.480
to find the sum of
the first 100 numbers.

00:00:31.480 --> 00:00:35.350
So that would be 1
plus 2 plus 3 plus,

00:00:35.350 --> 00:00:40.410
and you would go all
the way to 99 plus 100.

00:00:40.410 --> 00:00:45.080
So mathematicians said, well,
let's find some notation,

00:00:45.080 --> 00:00:47.330
instead of having to do this
dot dot dot thing-- which

00:00:47.330 --> 00:00:50.090
you will see sometimes
done-- so that we can more

00:00:50.090 --> 00:00:52.640
cleanly express
these types of sums.

00:00:52.640 --> 00:00:54.980
And that's where Sigma
notation comes from.

00:00:54.980 --> 00:00:58.140
So this sum up here, right
over here, this first one,

00:00:58.140 --> 00:01:01.490
it could be
represented as Sigma.

00:01:01.490 --> 00:01:04.780
Use a capital Sigma, this
Greek letter right over here.

00:01:04.780 --> 00:01:06.840
And what you do is
you define an index.

00:01:06.840 --> 00:01:10.080
And you could start your
index at some value.

00:01:10.080 --> 00:01:12.650
So let's say your
index starts at 1.

00:01:12.650 --> 00:01:14.660
I'll just use i for index.

00:01:14.660 --> 00:01:20.830
So let's say that i starts at
1, and I'm going to go to 10.

00:01:20.830 --> 00:01:23.690
So i starts at 1,
and it goes to 10.

00:01:23.690 --> 00:01:26.390
And I'm going to sum up the i's.

00:01:26.390 --> 00:01:29.920
So how does this translate
into this right over here?

00:01:29.920 --> 00:01:32.650
Well, what you do is you
start wherever the index is.

00:01:32.650 --> 00:01:35.980
If the index is at
1, set i equal to 1.

00:01:35.980 --> 00:01:39.560
Write the 1 down, and then
you increment the index.

00:01:39.560 --> 00:01:42.386
And so i will then
be equal to 2.

00:01:42.386 --> 00:01:43.560
i is 2.

00:01:43.560 --> 00:01:44.390
Put the 2 down.

00:01:44.390 --> 00:01:47.290
And you're summing each
of these terms as you go.

00:01:47.290 --> 00:01:50.320
And you go all the way
until i is equal to 10.

00:01:53.170 --> 00:01:54.920
So given what I just
told you, I encourage

00:01:54.920 --> 00:01:58.230
you to pause this video and
write the Sigma notation

00:01:58.230 --> 00:02:01.580
for this sum right over here.

00:02:01.580 --> 00:02:03.220
Assuming you've
given a go at it,

00:02:03.220 --> 00:02:05.125
well, this would be the sum.

00:02:05.125 --> 00:02:06.500
The first term,
well, it might be

00:02:06.500 --> 00:02:08.800
easy to just say we'll
start at i equals 1 again.

00:02:11.590 --> 00:02:15.000
But now we're not going to
stop until i equals 100,

00:02:15.000 --> 00:02:19.380
and we're going to
sum up all of the i's.

00:02:19.380 --> 00:02:20.820
Let's do another example.

00:02:20.820 --> 00:02:36.620
Let's imagine the sum from
i equals 0 to 50 of-- I

00:02:36.620 --> 00:02:40.163
don't know, let me
say-- pi i squared.

00:02:43.180 --> 00:02:44.430
What would this sum look like?

00:02:44.430 --> 00:02:46.513
And once again, I encourage
you to pause the video

00:02:46.513 --> 00:02:50.090
and write it out,
expand out this sum.

00:02:50.090 --> 00:02:52.920
Well, let's just
go step by step.

00:02:52.920 --> 00:02:56.400
When i equals 0, this will
be pi times 0 squared.

00:02:56.400 --> 00:02:58.250
And that's clearly 0,
but I'll write it out.

00:02:58.250 --> 00:03:02.330
pi times 0 squared.

00:03:02.330 --> 00:03:03.869
Then we increase our i.

00:03:03.869 --> 00:03:05.910
And, well, we make sure
that we haven't hit this,

00:03:05.910 --> 00:03:08.370
that our i isn't already
this top boundary

00:03:08.370 --> 00:03:10.420
right over here
or this top value.

00:03:10.420 --> 00:03:13.530
So now we said i
equals 1, pi times 1

00:03:13.530 --> 00:03:20.620
squared-- so plus
pi times 1 squared.

00:03:24.080 --> 00:03:26.990
Well, is 1 our top value right
over here, where we stop?

00:03:26.990 --> 00:03:27.490
No.

00:03:27.490 --> 00:03:28.670
So we keep going.

00:03:28.670 --> 00:03:31.840
So then we go i
equals 2, pi times 2

00:03:31.840 --> 00:03:37.855
squared-- so plus
pi times 2 squared.

00:03:40.610 --> 00:03:42.240
I think you see
the pattern here.

00:03:42.240 --> 00:03:44.890
And we're just going to
keep going all the way

00:03:44.890 --> 00:03:47.650
until, at some point-- we're
going to keeping incrementing

00:03:47.650 --> 00:03:49.280
our i. i is going to be 49.

00:03:49.280 --> 00:03:51.660
So it's going to be
pi times 49 squared.

00:03:55.210 --> 00:03:58.900
And then finally we increment
i. i equal becomes 50,

00:03:58.900 --> 00:04:05.710
and so we're going to have
plus pi times 50 squared.

00:04:05.710 --> 00:04:08.010
And then we say,
OK, our i is finally

00:04:08.010 --> 00:04:11.750
equal to this top boundary,
and now we can stop.

00:04:11.750 --> 00:04:13.640
And so you can
see this notation,

00:04:13.640 --> 00:04:18.360
this Sigma notation for this
sum was a much cleaner way,

00:04:18.360 --> 00:04:20.519
a much purer way,
of representing this

00:04:20.519 --> 00:04:22.410
than having to write
out the entire sum.

00:04:22.410 --> 00:04:26.510
But you'll see people switch
back and forth between the two.


Geometric sequence or progression

WEBVTT
Kind: captions
Language: en

00:00:00.697 --> 00:00:02.516
- [Instructor] I'm going
to construct a sequence.

00:00:02.516 --> 00:00:04.275
We're going to start with some number.

00:00:04.275 --> 00:00:05.993
Let's say I start with the number, a.

00:00:05.993 --> 00:00:08.297
And then each successive
term of the sequence,

00:00:08.297 --> 00:00:09.880
I'm going to multiply the,

00:00:09.880 --> 00:00:11.874
to get each successive
term of the sequence,

00:00:11.874 --> 00:00:13.664
I'm going to muliply the previous term

00:00:13.664 --> 00:00:16.243
by some fixed non-zero number,

00:00:16.243 --> 00:00:17.525
and I'm going to call that r.

00:00:17.525 --> 00:00:19.760
So the next term is going to be a,

00:00:19.760 --> 00:00:22.561
is going to be a times r.

00:00:22.561 --> 00:00:24.265
And then the term after that,

00:00:24.265 --> 00:00:27.154
I'm going to multipy this thing times r.

00:00:27.154 --> 00:00:28.740
So it's going to be a,

00:00:28.740 --> 00:00:30.931
if you multiply ar times r,

00:00:30.931 --> 00:00:33.431
that's going to be ar-squared.

00:00:34.716 --> 00:00:38.022
And then if you were going to
multiply this term times r,

00:00:38.022 --> 00:00:41.605
you would get a times
r to the third power.

00:00:43.221 --> 00:00:46.514
And you could keep going
on and on and on and on.

00:00:46.514 --> 00:00:49.282
And this type of sequence
or this type of progression

00:00:49.282 --> 00:00:51.240
is called a geometric.

00:00:51.240 --> 00:00:54.073
Geometric Sequence or Progression.

00:00:55.865 --> 00:00:57.448
Geometric Sequence.

00:00:58.304 --> 00:01:00.769
You start with some first value.

00:01:00.769 --> 00:01:02.163
Let me circle that in a different color,

00:01:02.163 --> 00:01:03.510
since I already used the green.

00:01:03.510 --> 00:01:05.822
So you start with some first value,

00:01:05.822 --> 00:01:08.508
and then to get each successive term,

00:01:08.508 --> 00:01:10.910
you multiply by this fixed number.

00:01:10.910 --> 00:01:13.305
In this case, this fixed number is r.

00:01:13.305 --> 00:01:16.722
And so we call r, our common ratio.

00:01:16.722 --> 00:01:18.139
Our common ratio.

00:01:19.404 --> 00:01:21.246
Why is it called a common ratio?

00:01:21.246 --> 00:01:23.542
Well take any two successive terms.

00:01:23.542 --> 00:01:25.274
Take this term and this term,

00:01:25.274 --> 00:01:28.780
and divide this term by
this term right over here,

00:01:28.780 --> 00:01:31.738
ar to the third, divided by ar-squared.

00:01:31.738 --> 00:01:34.534
So if you find the ratio
between these two things.

00:01:34.534 --> 00:01:36.814
Let me rewrite this in the same colors.

00:01:36.814 --> 00:01:40.524
So if you took ar to the third power,

00:01:40.524 --> 00:01:43.239
and were to divide it
by the term before it.

00:01:43.239 --> 00:01:47.545
So if you were to divide it by ar-squared,

00:01:47.545 --> 00:01:49.373
what are you going to be left with?

00:01:49.373 --> 00:01:51.528
Well a divided by a is one,

00:01:51.528 --> 00:01:55.749
r to the third divided by
r-squared is just going to be r.

00:01:55.749 --> 00:01:57.624
And this is true if you divide any term

00:01:57.624 --> 00:01:59.189
by the term before it.

00:01:59.189 --> 00:02:02.056
Or if you find the ratio between any term

00:02:02.056 --> 00:02:02.994
and the term before it,

00:02:02.994 --> 00:02:04.528
it's going to be r.

00:02:04.528 --> 00:02:07.017
And so that's why it's
called a common ratio.

00:02:07.017 --> 00:02:11.245
And so let's look at some
examples of geometric sequences.

00:02:11.245 --> 00:02:13.803
So if I start with the number,

00:02:13.803 --> 00:02:17.932
if I start with the number
five, so my a is five

00:02:17.932 --> 00:02:20.249
and then each time I'm going to multiply,

00:02:20.249 --> 00:02:23.341
I'm going to multiply by, I don't know.

00:02:23.341 --> 00:02:25.517
Let's say I multiply by 1/7.

00:02:25.517 --> 00:02:29.684
So then the next term is
going to be five over seven.

00:02:32.132 --> 00:02:34.026
What's the next term going to be?

00:02:34.026 --> 00:02:37.006
Well, I'm going to multiply
this thing times 1/7.

00:02:37.006 --> 00:02:41.237
So that's going to be
5/7 times 1/7 is 5/49.

00:02:41.237 --> 00:02:45.404
So it's going to be five
over seven-squared or 49.

00:02:47.173 --> 00:02:49.982
If I were to multiply this times 1/7,

00:02:49.982 --> 00:02:51.511
what am I going to get?

00:02:51.511 --> 00:02:52.536
I'll just change the notation.

00:02:52.536 --> 00:02:54.203
I'll get five times,

00:02:55.217 --> 00:02:56.279
I don't actually know in my head

00:02:56.279 --> 00:02:57.786
what seven to the third power is,

00:02:57.786 --> 00:03:01.286
I guess I could calculate it, 280 plus 63,

00:03:03.011 --> 00:03:04.266
let's see, so that would be

00:03:04.266 --> 00:03:06.552
let's see seven times 40 is 280,

00:03:06.552 --> 00:03:08.469
seven times nine is 63,

00:03:09.478 --> 00:03:13.667
so you're going to get to 343, I believe.

00:03:13.667 --> 00:03:15.281
Let me see, did I do that right?

00:03:15.281 --> 00:03:17.255
Seven times 280,

00:03:17.255 --> 00:03:18.505
plus 63 is 343,

00:03:20.091 --> 00:03:22.271
and you can just keep going.

00:03:22.271 --> 00:03:24.521
So this right over here is an example

00:03:24.521 --> 00:03:25.771
of a geometric,

00:03:26.775 --> 00:03:28.764
of a geometric sequence.

00:03:28.764 --> 00:03:30.272
Started with some first value

00:03:30.272 --> 00:03:33.274
and each successive value,
I multiplied by 1/7.

00:03:33.274 --> 00:03:36.018
1/7 is the common ratio here.

00:03:36.018 --> 00:03:38.417
Let me give you another one.

00:03:38.417 --> 00:03:40.129
Let's say I have this,

00:03:40.129 --> 00:03:45.099
let's say I have three and
then let's say I have six,

00:03:45.099 --> 00:03:47.099
and let's say I have 12.

00:03:48.472 --> 00:03:49.722
Then I have 24,

00:03:51.520 --> 00:03:52.770
then I have 48.

00:03:53.902 --> 00:03:55.537
Is this a geometric series?

00:03:55.537 --> 00:03:59.511
And if it is, what is
the common ratio here?

00:03:59.511 --> 00:04:01.761
Well you could figure out the common ratio

00:04:01.761 --> 00:04:04.767
by just taking any two
successive terms and dividing.

00:04:04.767 --> 00:04:07.782
Well first, you could
try it with two terms.

00:04:07.782 --> 00:04:10.262
A 12 divided by six is two.

00:04:10.262 --> 00:04:12.047
So we have to multiply by two.

00:04:12.047 --> 00:04:15.489
To go from 12 to 24, you multipled by two.

00:04:15.489 --> 00:04:19.003
To go from 24 to 48, you
have to multiply by two.

00:04:19.003 --> 00:04:21.517
To go from three to six,
we have to multiply by two.

00:04:21.517 --> 00:04:22.742
So you get a fixed common ratio.

00:04:22.742 --> 00:04:25.257
For any of these terms,
if we multiplied by two

00:04:25.257 --> 00:04:27.249
and say multiplied by three,

00:04:27.249 --> 00:04:29.516
and so we didn't multiply
by the same thing,

00:04:29.516 --> 00:04:33.599
then it wouldn't be a
geometric sequence anymore.

00:04:34.567 --> 00:04:36.277
So this clearly is a geometric sequence,

00:04:36.277 --> 00:04:38.008
I forgot to mention that.


Geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.850
In the last video we saw
that a geometric progression,

00:00:02.850 --> 00:00:05.340
or a geometric sequence,
is just a sequence where

00:00:05.340 --> 00:00:08.700
each successive term is the
previous term multiplied

00:00:08.700 --> 00:00:10.250
by a fixed value.

00:00:10.250 --> 00:00:12.310
And we call that fixed
value the common ratio.

00:00:12.310 --> 00:00:14.430
So, for example, in this
sequence right over here,

00:00:14.430 --> 00:00:18.360
each term is the previous
term multiplied by 2.

00:00:18.360 --> 00:00:21.230
So 2 is our common ratio.

00:00:21.230 --> 00:00:26.050
And any non-zero value
can be our common ratio.

00:00:26.050 --> 00:00:27.450
It can even be a negative value.

00:00:27.450 --> 00:00:31.230
So, for example, you could
have a geometric sequence

00:00:31.230 --> 00:00:32.170
that looks like this.

00:00:32.170 --> 00:00:37.450
Maybe start at one, and
maybe our common ratio,

00:00:37.450 --> 00:00:39.370
let's say it's negative 3.

00:00:39.370 --> 00:00:41.860
So 1 times negative
3 is negative 3.

00:00:41.860 --> 00:00:45.240
Negative 3 times
negative 3 is positive 9.

00:00:45.240 --> 00:00:49.510
Positive 9 times negative
3 is negative 27.

00:00:49.510 --> 00:00:53.870
And then negative 27 times
negative 3 is positive 81.

00:00:53.870 --> 00:00:56.500
And you could keep
going on and on and on.

00:00:56.500 --> 00:00:58.410
What I now want to
focus on in this video

00:00:58.410 --> 00:01:00.960
is the sum of a
geometric progression

00:01:00.960 --> 00:01:03.820
or a geometric sequence,
and we would call that

00:01:03.820 --> 00:01:06.150
a geometric series.

00:01:06.150 --> 00:01:08.130
Let's scroll down a little bit.

00:01:08.130 --> 00:01:15.280
So now we're going to talk
about geometric series, which

00:01:15.280 --> 00:01:17.750
is really just the sum
of a geometric sequence.

00:01:17.750 --> 00:01:20.360
So, for example,
a geometric series

00:01:20.360 --> 00:01:22.390
would just be a sum
of this sequence.

00:01:22.390 --> 00:01:31.540
So if we just said
1 plus negative 3,

00:01:31.540 --> 00:01:36.890
plus 9, plus
negative 27, plus 81,

00:01:36.890 --> 00:01:38.890
and we were to go
on, and on, and on,

00:01:38.890 --> 00:01:41.484
this would be a
geometric series.

00:01:41.484 --> 00:01:43.150
And we could do it
with this one up here

00:01:43.150 --> 00:01:45.191
just to really make it
clear of what we're doing.

00:01:45.191 --> 00:01:51.470
So if we said 3 plus 6,
plus 12, plus 24, plus 48,

00:01:51.470 --> 00:01:54.800
this once again is a
geometric series, just

00:01:54.800 --> 00:01:59.450
the sum of a geometric sequence
or a geometric progression.

00:01:59.450 --> 00:02:01.940
So how would we represent this
in general terms and maybe

00:02:01.940 --> 00:02:03.820
using sigma notation?

00:02:03.820 --> 00:02:06.470
Well, we'll start with
whatever our first term is.

00:02:06.470 --> 00:02:08.710
And over here if we want
to speak in general terms

00:02:08.710 --> 00:02:11.990
we could call that
a, our first term.

00:02:11.990 --> 00:02:17.350
So we'll start with our
first term, a, and then

00:02:17.350 --> 00:02:19.430
each successive term
that we're going to add

00:02:19.430 --> 00:02:22.730
is going to be a times
our common ratio.

00:02:22.730 --> 00:02:24.610
And we'll call that
common ratio r.

00:02:24.610 --> 00:02:26.899
So the second term is a times r.

00:02:26.899 --> 00:02:28.440
Then the third term,
we're just going

00:02:28.440 --> 00:02:30.040
to multiply this one times r.

00:02:30.040 --> 00:02:32.275
So it's going to be
a times r squared.

00:02:35.810 --> 00:02:43.330
And then we can keep going, plus
a times r to the third power.

00:02:43.330 --> 00:02:46.190
And let's say we're going to
do a finite geometric series.

00:02:46.190 --> 00:02:48.500
So we're not going to just
keep on going forever.

00:02:48.500 --> 00:02:50.620
Let's say we keep
going all the way

00:02:50.620 --> 00:02:58.066
until we get to some
a times r to the n.

00:02:58.066 --> 00:03:03.200
a times r to the n-th power.

00:03:03.200 --> 00:03:05.540
So how can we represent
this with sigma notation?

00:03:05.540 --> 00:03:09.460
And I encourage you to pause the
video and try it on your own.

00:03:09.460 --> 00:03:13.977
Well, we could think
about it this way.

00:03:13.977 --> 00:03:15.310
And I'll give you a little hint.

00:03:15.310 --> 00:03:17.701
You could view this term
right over here as a times r

00:03:17.701 --> 00:03:18.200
to the 0.

00:03:18.200 --> 00:03:20.140
And let me write it down.

00:03:20.140 --> 00:03:21.920
This is a times r to the 0.

00:03:21.920 --> 00:03:24.180
This is a times r to
the first, r squared,

00:03:24.180 --> 00:03:27.540
r third, and now the pattern
might be emerging for you.

00:03:27.540 --> 00:03:32.120
So we can write this as
the sum, so capital sigma

00:03:32.120 --> 00:03:33.170
right over here.

00:03:33.170 --> 00:03:35.740
We can start our index at 0.

00:03:35.740 --> 00:03:50.370
So we could say from k
equals 0 all the way to k

00:03:50.370 --> 00:04:10.700
equals n of a times
r to the k-th power.

00:04:13.510 --> 00:04:15.070
And so this is,
using sigma notation,

00:04:15.070 --> 00:04:19.290
a general way to represent
a geometric series where

00:04:19.290 --> 00:04:23.660
r is some non-zero common ratio.

00:04:23.660 --> 00:04:26.550
It can even be a negative value.


Another derivation of the sum of an infinite geometric series

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.700
What I want to do is
another "proofy-like"

00:00:02.700 --> 00:00:07.550
thing to think about the sum of
an infinite geometric series.

00:00:07.550 --> 00:00:09.310
And we'll use a
very similar idea

00:00:09.310 --> 00:00:13.820
to what we used to find the sum
of a finite geometric series.

00:00:13.820 --> 00:00:16.890
So let's say I have a geometric
series, an infinite geometric

00:00:16.890 --> 00:00:17.810
series.

00:00:17.810 --> 00:00:20.000
So we're going to
start at k equals 0,

00:00:20.000 --> 00:00:22.040
and we're never going to stop.

00:00:22.040 --> 00:00:24.880
We're it's going all
the way to infinity.

00:00:24.880 --> 00:00:27.480
So we're never going to
stop adding terms here.

00:00:27.480 --> 00:00:32.330
And it's going to be our first
term times our common ratio.

00:00:32.330 --> 00:00:35.770
Our common ratio
to the kth power.

00:00:35.770 --> 00:00:37.680
Actually let me do
k and that color. k

00:00:37.680 --> 00:00:41.900
equals 0 all the
way to infinity.

00:00:41.900 --> 00:00:46.000
And so let's just call
this thing right over here,

00:00:46.000 --> 00:00:52.790
let's call this s sub infinity.

00:00:52.790 --> 00:00:56.790
We're going all the way to
infinity right over here.

00:00:56.790 --> 00:00:58.530
And so this, if we
were to expand it out

00:00:58.530 --> 00:01:04.129
is going to be equal to a
times r to the 0-- actually

00:01:04.129 --> 00:01:07.710
let me just write it out
like that which is just a.

00:01:07.710 --> 00:01:18.690
a times r to the 0 power plus
a times r to the 1st power.

00:01:18.690 --> 00:01:21.410
r to the 1st power.

00:01:21.410 --> 00:01:26.940
Plus a times r to the 2nd power.

00:01:26.940 --> 00:01:29.850
r to the 2nd power.

00:01:29.850 --> 00:01:35.510
Plus-- and we could just
keep going on and on and on.

00:01:35.510 --> 00:01:37.940
I think you get
the general idea.

00:01:37.940 --> 00:01:39.890
Now just like when
we tried to derive

00:01:39.890 --> 00:01:43.610
a formula for the sum of
a finite geometric series

00:01:43.610 --> 00:01:45.900
we just said, well what
happens if you take the sum

00:01:45.900 --> 00:01:48.000
and if you were to
multiply every term

00:01:48.000 --> 00:01:49.150
by your common ratio.

00:01:49.150 --> 00:01:50.340
Every term by r.

00:01:50.340 --> 00:01:51.550
So let's do that.

00:01:51.550 --> 00:01:52.980
Let's imagine this sum.

00:01:52.980 --> 00:01:55.866
And we're going to
multiply every term by r.

00:01:55.866 --> 00:01:57.240
And the reason
why I said this is

00:01:57.240 --> 00:02:01.750
"proofy" is this is not always
clear-- It's a little bit,

00:02:01.750 --> 00:02:04.150
when you're multiplying
something times infinite terms

00:02:04.150 --> 00:02:06.760
or an infinite sum, at
least this will be at least

00:02:06.760 --> 00:02:08.343
give you the general idea.

00:02:08.343 --> 00:02:10.259
Or when you start thinking
about and infinity,

00:02:10.259 --> 00:02:12.770
sometimes I just think about
things a little bit deeper.

00:02:12.770 --> 00:02:16.370
So r times this infinite sum?

00:02:16.370 --> 00:02:18.400
Well that's going to
be equal to-- We're

00:02:18.400 --> 00:02:21.520
just going to multiply
every term here times r.

00:02:21.520 --> 00:02:29.780
So a r to the 0'th power times
r is going to be a times r.

00:02:29.780 --> 00:02:32.730
a times r to the 1st power.

00:02:32.730 --> 00:02:34.250
Multiply this one times r?

00:02:34.250 --> 00:02:38.940
You're going to get a
times r to the 2nd power.

00:02:38.940 --> 00:02:40.430
a times r to the 2nd power.

00:02:40.430 --> 00:02:42.110
I think you see
where this is going.

00:02:42.110 --> 00:02:43.540
Multiply this one times r?

00:02:43.540 --> 00:02:50.160
You're going to get plus a
times r to the 3rd power.

00:02:50.160 --> 00:02:53.110
And we would just keep on going.

00:02:53.110 --> 00:02:55.630
We'd just keep on going.

00:02:55.630 --> 00:02:56.910
So let me just show that.

00:02:56.910 --> 00:02:59.236
So plus dot dot dot.

00:02:59.236 --> 00:03:01.210
Now what happens if
we were to subtract

00:03:01.210 --> 00:03:03.704
this sum from this top sum?

00:03:03.704 --> 00:03:05.120
So on the left
hand side, we could

00:03:05.120 --> 00:03:13.930
express that as our sum s sub
infinity minus our common ratio

00:03:13.930 --> 00:03:17.300
times s sub infinity.

00:03:17.300 --> 00:03:19.710
Is going to be equal to--

00:03:19.710 --> 00:03:22.480
So when you subtract you're
going to have a times r

00:03:22.480 --> 00:03:24.730
to the 0'th power, which is
really just the same thing

00:03:24.730 --> 00:03:25.656
as a.

00:03:25.656 --> 00:03:27.800
That's just going to be
a. a times r to the 0

00:03:27.800 --> 00:03:29.745
is just a times 1 which is a.

00:03:29.745 --> 00:03:32.050
We'll write in that same color.

00:03:32.050 --> 00:03:33.260
Is equal to a.

00:03:33.260 --> 00:03:35.060
But every other
term, you're going

00:03:35.060 --> 00:03:36.833
to have a times r to
the 1st, but you're

00:03:36.833 --> 00:03:38.972
gonna subtract a
times r to the 1st.

00:03:38.972 --> 00:03:40.680
You have a times r to
the 2nd, but you're

00:03:40.680 --> 00:03:42.660
going to subtract a
times r to the 2nd.

00:03:42.660 --> 00:03:47.190
So every other term is
going to be subtracted away.

00:03:47.190 --> 00:03:51.320
And this happens all
the way to infinity.

00:03:51.320 --> 00:03:52.695
It never, never ends.

00:03:52.695 --> 00:03:54.070
So the only term
that you're left

00:03:54.070 --> 00:03:57.150
with is just that
first one, is just a.

00:03:57.150 --> 00:03:59.930
And so now we can actually
try to solve for our sum.

00:03:59.930 --> 00:04:03.530
If you factor out
the s sub infinity,

00:04:03.530 --> 00:04:07.830
you are left with 1 minus r.

00:04:07.830 --> 00:04:09.726
1 minus r.

00:04:09.726 --> 00:04:13.195
s times s, our sum, times
1 minus r is equal to a.

00:04:13.195 --> 00:04:16.070
Divide both sides
by 1 minus r, and we

00:04:16.070 --> 00:04:18.667
get that our sum, the
thing that we cared about--

00:04:18.667 --> 00:04:20.750
And once again, this is
kind of an amazing result.

00:04:20.750 --> 00:04:23.980
That we're taking the sum of
an infinite number of terms

00:04:23.980 --> 00:04:26.540
and under the
proper constraints,

00:04:26.540 --> 00:04:30.760
we are going to
get a finite value.

00:04:30.760 --> 00:04:40.650
So this is going to be
equal to a over 1 minus r.

00:04:40.650 --> 00:04:42.740
So once again,
it's kind of neat.

00:04:42.740 --> 00:04:49.620
If let's say I had the sum,
let's say we started with 5,

00:04:49.620 --> 00:04:53.850
and then each time we
were to multiply by 3/5.

00:04:53.850 --> 00:05:03.660
So 5 plus 3/5 times 5 is 3,
times 3/5 is going to be 9/5.

00:05:03.660 --> 00:05:12.360
9/5, or I'll multiply by 9/5
again-- Oh, sorry not 9/5.

00:05:12.360 --> 00:05:13.760
My brain isn't working right!

00:05:13.760 --> 00:05:19.330
5 times 3/5 is going
to be 3 times 3/5.

00:05:19.330 --> 00:05:23.010
Is going to be-- 3
times this is going

00:05:23.010 --> 00:05:24.825
to be 9/5-- actually
that was right.

00:05:24.825 --> 00:05:26.070
My brain is working right.

00:05:26.070 --> 00:05:31.710
Times 3/5 is going
to be 27 over 25.

00:05:31.710 --> 00:05:38.040
Times 3/5 is going to be 81/125.

00:05:38.040 --> 00:05:40.432
And we keep on going on
and on and on forever.

00:05:40.432 --> 00:05:41.890
And notice these
terms are starting

00:05:41.890 --> 00:05:43.640
to get smaller and
smaller and smaller.

00:05:43.640 --> 00:05:44.820
Well actually all
of them are getting

00:05:44.820 --> 00:05:46.153
smaller and smaller and smaller.

00:05:46.153 --> 00:05:47.860
We're multiplying
by 3/5 every time.

00:05:47.860 --> 00:05:49.627
We now know what the
sum is going to be.

00:05:49.627 --> 00:05:51.210
It's going to be our
first term-- it's

00:05:51.210 --> 00:05:54.730
going to be 5-- over 1
minus our common ratio.

00:05:54.730 --> 00:05:57.960
And our common ratio
in this case is 3/5.

00:05:57.960 --> 00:06:03.900
So this is going to be
equal to 5 over 2/5,

00:06:03.900 --> 00:06:11.210
which is the same
thing as 5 times 5/2

00:06:11.210 --> 00:06:17.570
which is 25/2 which is equal
to 12 and 1/2, or 12.5.

00:06:17.570 --> 00:06:19.550
Once again, amazing result.

00:06:19.550 --> 00:06:22.990
I'm taking a sum of
infinite terms here,

00:06:22.990 --> 00:06:25.710
and I was able to
get a finite result.

00:06:25.710 --> 00:06:28.670
And once again, when
does this happen?

00:06:28.670 --> 00:06:32.130
Well, if our common ratio--
if the absolute value

00:06:32.130 --> 00:06:34.945
of our common ratio--
is less than 1,

00:06:34.945 --> 00:06:36.320
then these terms
are going to get

00:06:36.320 --> 00:06:37.653
smaller and smaller and smaller.

00:06:37.653 --> 00:06:41.160
And you'll even see here it
even works out mathematically

00:06:41.160 --> 00:06:43.260
in this denominator
that you are going

00:06:43.260 --> 00:06:45.050
to get a reasonable answer.

00:06:45.050 --> 00:06:46.709
And it makes sense
because these terms

00:06:46.709 --> 00:06:48.500
are getting smaller
and smaller and smaller

00:06:48.500 --> 00:06:50.000
that this thing will converge.

00:06:50.000 --> 00:06:50.880
Even if r is 0.

00:06:50.880 --> 00:06:54.850
If r is 0, we're still
not dealing strictly

00:06:54.850 --> 00:06:58.440
with a geometric series anymore,
but obviously if r was 0,

00:06:58.440 --> 00:07:01.470
then you're really only going
to have this-- well, even

00:07:01.470 --> 00:07:03.140
this first term is
kind of under debate

00:07:03.140 --> 00:07:05.770
depending on how you
define what 0 to 0 is.

00:07:05.770 --> 00:07:07.770
But if your first term
you just said would be a,

00:07:07.770 --> 00:07:11.170
then clearly you'd just be left
with a is the sum, and a over 1

00:07:11.170 --> 00:07:12.330
minus 0 is still a.

00:07:12.330 --> 00:07:16.280
So this formula that we just
derived does hold up for that.

00:07:16.280 --> 00:07:20.300
It does start to break down if
r is equal to 1 or negative 1.

00:07:20.300 --> 00:07:22.730
If r is equal to 1 then
as you imagine here,

00:07:22.730 --> 00:07:25.370
you just have a plus
a plus a plus a,

00:07:25.370 --> 00:07:26.720
going on and on forever.

00:07:26.720 --> 00:07:29.120
If r is equal to negative 1
you just keep oscillating.

00:07:29.120 --> 00:07:33.800
a, minus a, plus a, minus a.

00:07:33.800 --> 00:07:37.300
And so the sum's value keeps
oscillating between two values.

00:07:37.300 --> 00:07:40.240
So in general this
infinite geometric series

00:07:40.240 --> 00:07:42.990
is going to converge
if the absolute value

00:07:42.990 --> 00:07:46.030
of your common ratio
is less than 1.

00:07:46.030 --> 00:07:49.860
Or another way of saying
that, if your common ratio is

00:07:49.860 --> 00:07:53.870
between 1 and negative 1.