Finding particular linear solution to differential equation

WEBVTT
Kind: captions
Language: en

00:00:00.241 --> 00:00:02.159
- So let's get a little bit more comfort

00:00:02.159 --> 00:00:03.525
in our understanding of what

00:00:03.525 --> 00:00:05.904
a differential equation even is.

00:00:06.277 --> 00:00:08.518
So here we have a differential equation.

00:00:08.518 --> 00:00:11.126
We haven't started exploring how we find

00:00:11.126 --> 00:00:13.611
the solutions for a
differential equations yet.

00:00:13.611 --> 00:00:15.132
But let's just say you saw this,

00:00:15.132 --> 00:00:16.958
and someone just walked
up to you on the street

00:00:16.958 --> 00:00:18.964
and says, "Hey, I will give you a clue,

00:00:19.558 --> 00:00:21.764
"that there's a solution to
this differential equation

00:00:21.764 --> 00:00:23.733
"that is essentially a linear function,

00:00:23.733 --> 00:00:26.801
"where y is equal to mx plus b,

00:00:27.413 --> 00:00:28.420
"and you just need to figure out

00:00:28.042 --> 00:00:31.300
"the m's and the b's,
or maybe the m and the b

00:00:31.678 --> 00:00:35.054
"that makes this linear function

00:00:35.054 --> 00:00:37.592
"satisfy this differential equation."

00:00:38.183 --> 00:00:40.098
What I now encourage you to do,

00:00:40.098 --> 00:00:43.007
is pause the video and
see if you can do it.

00:00:43.077 --> 00:00:44.084
So I'm assuming you have paused it,

00:00:45.047 --> 00:00:46.181
and had a go at it.

00:00:46.604 --> 00:00:47.640
So let's think this through together.

00:00:48.233 --> 00:00:50.247
If we know that this kind of a solution

00:00:50.373 --> 00:00:51.526
can be described in this way,

00:00:51.526 --> 00:00:52.597
we need to figure out
some m's and b's here.

00:00:53.236 --> 00:00:54.792
This is telling us that if we were to take

00:00:54.792 --> 00:00:56.986
the derivative of this with respect to x,

00:00:56.986 --> 00:00:59.048
if we take the derivative of
mx plus b with respect to x,

00:00:59.606 --> 00:01:01.639
that that should be equal
to negative 2 times x

00:01:01.936 --> 00:01:03.367
plus 3 times y.

00:01:03.367 --> 00:01:06.614
well, we know y is this thing, minus 5.

00:01:06.614 --> 00:01:08.846
And that should be true for all x's,

00:01:08.846 --> 00:01:10.843
in order for this to be a solution

00:01:10.843 --> 00:01:12.230
to this differential equation.

00:01:12.023 --> 00:01:13.318
Remember, the solution to
a differential equation

00:01:13.525 --> 00:01:15.359
is not a value or a set of values.

00:01:15.359 --> 00:01:19.202
It is a function or a set of functions.

00:01:19.202 --> 00:01:22.747
So in order for this to satisfy
this differential equation,

00:01:22.747 --> 00:01:25.463
it needs to be true for
all of these x's here.

00:01:25.463 --> 00:01:26.679
So let's work through it.

00:01:26.679 --> 00:01:30.126
Let's figure out first what our dy dx is.

00:01:30.126 --> 00:01:31.839
So dy dx.

00:01:32.247 --> 00:01:34.533
We'll just take the derivative
here with respect to x

00:01:34.533 --> 00:01:37.676
dy dx is derivative of
mx with respect to x,

00:01:37.676 --> 00:01:39.448
is just going to be m.

00:01:39.448 --> 00:01:41.561
And of course derivative
of b with respect to x,

00:01:41.561 --> 00:01:42.694
just a constant,

00:01:42.694 --> 00:01:43.757
so it's just going to be zero.

00:01:44.324 --> 00:01:46.951
So dy dx is m.

00:01:46.951 --> 00:01:51.243
So we could write m is
equal to negative 2x,

00:01:51.243 --> 00:01:54.339
is equal to negative 2x,

00:01:54.339 --> 00:01:57.378
plus 3 times, and instead
of putting y there

00:01:57.729 --> 00:01:58.798
I could write mx plus b.

00:01:59.419 --> 00:02:02.112
Remember y is equal to mx plus b.

00:02:02.112 --> 00:02:03.723
And just as a repeated reminder,

00:02:03.723 --> 00:02:05.735
this has to be true for all x's.

00:02:05.843 --> 00:02:10.842
mx plus b, and then of
course we have the minus 5.

00:02:11.973 --> 00:02:14.546
So if you weren't able to
solve it the first time,

00:02:14.546 --> 00:02:16.608
I encourage you to start from here,

00:02:16.608 --> 00:02:20.791
and now figure out what
m and b needs to be

00:02:20.791 --> 00:02:23.842
in order for this
equation right over here,

00:02:23.842 --> 00:02:27.501
in order for this to be true for all x's.

00:02:27.501 --> 00:02:29.974
In order for this to be true for all x's.

00:02:29.974 --> 00:02:32.262
So assuming you have paused again

00:02:32.262 --> 00:02:33.899
and had a go at it, let's just keep

00:02:33.899 --> 00:02:35.338
algebraically manipulating this.

00:02:35.338 --> 00:02:37.601
I'll just switch to one color here.

00:02:37.787 --> 00:02:39.643
So we have m.

00:02:40.368 --> 00:02:42.437
m is equal to negative 2x plus,

00:02:43.058 --> 00:02:45.704
if we distribute this
3 we're going to have

00:02:45.704 --> 00:02:50.704
3mx plus 3b, and then of course
we're going to have minus 5.

00:02:51.065 --> 00:02:53.630
And now we can group the x terms.

00:02:54.215 --> 00:02:57.808
So if we were to group,
if we were to group ...

00:02:57.096 --> 00:02:59.111
Let me find a new color
here, maybe this blue.

00:03:00.135 --> 00:03:02.456
So if we were to take these
two and add them together

00:03:02.456 --> 00:03:05.955
that's going to be
negative 2 plus 3m times x,

00:03:05.955 --> 00:03:10.955
or we could write this
as 3m minus 2 times x,

00:03:11.319 --> 00:03:13.675
and then you have your constant terms,

00:03:13.675 --> 00:03:15.161
so you have these terms right over here,

00:03:15.161 --> 00:03:18.733
so plus 3b minus 5,

00:03:18.733 --> 00:03:20.985
and of course that's all
going to be equal to m.

00:03:20.985 --> 00:03:23.709
That's going to be equal to m.

00:03:23.709 --> 00:03:25.716
Now remember, this needs to be true,

00:03:25.716 --> 00:03:29.258
this needs to be true for all x's.

00:03:29.945 --> 00:03:31.592
So notice, over here I have

00:03:31.592 --> 00:03:35.110
some coefficient times x
on the right-hand side.

00:03:35.011 --> 00:03:38.024
But on the left-hand side I have no x's.

00:03:38.123 --> 00:03:40.759
So somehow, this thing must disappear.

00:03:40.759 --> 00:03:43.982
This is a constant, so
it's completely reasonable.

00:03:43.982 --> 00:03:46.005
It's completely reasonable
that this constant

00:03:46.212 --> 00:03:48.695
could be equal to m.

00:03:48.695 --> 00:03:51.609
But the only way that I
get these x's to disappear

00:03:51.609 --> 00:03:54.005
so all I'm left with is an m,

00:03:54.005 --> 00:03:56.507
is if this thing is equal to zero.

00:03:56.621 --> 00:03:57.584
Let me say that again,

00:03:57.584 --> 00:03:59.213
because I think it may be a little bit

00:03:59.213 --> 00:04:01.474
counter-intuitive what I'm about to do.

00:04:01.474 --> 00:04:04.306
We're saying that m, some constant value,

00:04:04.306 --> 00:04:06.485
is equal to some coefficient times x,

00:04:06.485 --> 00:04:08.822
plus some other constant value.

00:04:09.439 --> 00:04:10.898
Well, in order for a constant value

00:04:10.898 --> 00:04:13.146
to be equal to a coefficient times x

00:04:13.146 --> 00:04:14.550
plus some other constant value

00:04:14.055 --> 00:04:17.803
the coefficient on X
must be equal to zero.

00:04:18.538 --> 00:04:20.302
Another way to think about it is,

00:04:20.302 --> 00:04:21.885
this should be,

00:04:21.885 --> 00:04:23.684
you could rewrite the left-hand side here

00:04:23.684 --> 00:04:26.385
as zero x plus m.

00:04:26.385 --> 00:04:29.263
So you see, you kind of
match the coefficients.

00:04:29.263 --> 00:04:32.346
So zero must be equal to 3m minus 2,

00:04:32.346 --> 00:04:37.087
and m is equal to 3b, m
is equal to 3b minus 5.

00:04:37.739 --> 00:04:39.741
m is equal to 3b minus 5.

00:04:40.421 --> 00:04:43.795
So let's use that
knowledge, that information,

00:04:43.795 --> 00:04:45.346
to solve for m and b.

00:04:45.718 --> 00:04:47.320
So we could use this first one.

00:04:47.032 --> 00:04:50.058
So 3m minus 2 must be equal to zero.

00:04:50.346 --> 00:04:51.325
So let's write that.

00:04:51.325 --> 00:04:54.331
3m minus 2 is equal to zero,

00:04:54.385 --> 00:04:56.701
or 3m is equal to 2,

00:04:56.928 --> 00:04:59.459
or m is equal to 2/3.

00:04:59.459 --> 00:05:02.454
So we figured out what m is.

00:05:02.454 --> 00:05:04.832
Then we could use that information,

00:05:04.832 --> 00:05:06.444
because we know that,

00:05:07.214 --> 00:05:10.651
we know that m is equal to,

00:05:10.651 --> 00:05:13.652
is equal to 3b minus 5.

00:05:14.168 --> 00:05:19.168
m is 2/3, so we get 2/3
is equal to 3b minus 5.

00:05:19.892 --> 00:05:21.653
We could add 5 to both sides,

00:05:21.653 --> 00:05:25.529
which is the same thing as
adding 15/3 to both sides.

00:05:26.308 --> 00:05:27.323
Is that, did I do that right?

00:05:27.458 --> 00:05:28.917
Yeah, adding 5 to both sides

00:05:28.917 --> 00:05:31.203
is the same thing as
adding 15/3 to both sides.

00:05:31.203 --> 00:05:32.196
So let's do that.

00:05:32.196 --> 00:05:37.196
15/3 plus 15/3, these cancel out.

00:05:38.069 --> 00:05:39.184
That's just 5 right over there.

00:05:39.184 --> 00:05:41.501
On the left-hand side we have

00:05:41.501 --> 00:05:46.262
17/3 is equal to 3b,

00:05:46.262 --> 00:05:48.538
or if you divide both sides by 3

00:05:48.538 --> 00:05:52.370
you get b is equal to 17,

00:05:53.305 --> 00:05:58.305
b is equal to 17/9, and we're done.

00:05:59.458 --> 00:06:03.061
We just found a particular solution

00:06:03.061 --> 00:06:05.086
for this differential equation.

00:06:05.343 --> 00:06:10.343
The solution is y is
equal to 2/3x plus 17/9.

00:06:14.693 --> 00:06:16.949
And I encourage you,
after watching this video,

00:06:16.949 --> 00:06:19.236
to verify that this particular solution

00:06:19.236 --> 00:06:22.680
indeed does satisfy this
differential equation

00:06:22.068 --> 00:06:24.423
for all x's.

00:06:25.443 --> 00:06:27.979
For all x's.


Worked example: alternating series remainder

WEBVTT
Kind: captions
Language: en

00:00:00.606 --> 00:00:02.305
- [Voiceover] So we've got
an infinite series here.

00:00:02.305 --> 00:00:05.113
Negative one to the n plus
one over the square root of n.

00:00:05.113 --> 00:00:06.630
I always like to visualize this

00:00:06.630 --> 00:00:08.276
a little bit more, expand it out.

00:00:08.276 --> 00:00:10.685
So when n equals one the
first term is negative one

00:00:10.685 --> 00:00:12.461
to the second power, so
it's going to be positive.

00:00:12.461 --> 00:00:14.387
It's going to be one
over one and so it's one

00:00:14.387 --> 00:00:16.929
minus one over the square root of two

00:00:16.929 --> 00:00:19.575
plus one over the square root of three

00:00:19.575 --> 00:00:22.115
minus one over the square
root of four and then

00:00:22.115 --> 00:00:25.379
plus, minus, and we just go
on and on and on forever.

00:00:26.002 --> 00:00:28.810
Now when we looked at convergence tests

00:00:28.810 --> 00:00:32.218
for infinite series we
saw things like this.

00:00:32.235 --> 00:00:35.102
This passes the alternating series test

00:00:35.102 --> 00:00:37.015
and so we know that this converges.

00:00:37.015 --> 00:00:40.063
Let's say it converges to some value S.

00:00:40.473 --> 00:00:42.401
But what we're concerned
with in this video is

00:00:42.401 --> 00:00:44.923
not whether or not this
converges, but estimating

00:00:44.923 --> 00:00:47.436
what this actually converges to.

00:00:47.436 --> 00:00:51.197
We know that we can estimate
this by taking a partial sum.

00:00:51.827 --> 00:00:55.844
Let's say S sub k, this is the
first k terms right over here

00:00:55.844 --> 00:00:57.561
and then you're going to have a remainder,

00:00:57.561 --> 00:01:01.953
so plus the remainder after
you've taken the first k terms.

00:01:01.953 --> 00:01:04.882
So this actually starts
with the k plus first term.

00:01:05.283 --> 00:01:07.335
And so what I am concerned
about or what I want

00:01:07.335 --> 00:01:10.843
to figure out is what's the
minimum number of terms,

00:01:10.843 --> 00:01:15.030
what's the minimum k here
so that my remainder,

00:01:15.030 --> 00:01:17.666
so that the absolute value of my remainder

00:01:17.666 --> 00:01:22.666
is less than or equal to 0.001.

00:01:23.467 --> 00:01:25.164
And I encourage you to
pause the video based

00:01:25.164 --> 00:01:26.403
on what we've seen in previous

00:01:26.403 --> 00:01:28.981
alternating series estimation situations

00:01:28.981 --> 00:01:30.663
to see if you can figure this out.

00:01:31.620 --> 00:01:35.552
Alright I'm assuming
you've had a go at it.

00:01:35.552 --> 00:01:39.210
So let's just remind ourselves
what R sub k looks like.

00:01:39.210 --> 00:01:44.210
So R sub k is going to start
with the k plus first term.

00:01:46.211 --> 00:01:50.371
So it's going to be negative
one k plus one plus one,

00:01:50.371 --> 00:01:54.483
so it's negative one
to the k plus two over

00:01:54.483 --> 00:01:59.483
the square root of k plus one.

00:02:02.415 --> 00:02:06.343
And then the next term
I can just write plus

00:02:06.343 --> 00:02:10.465
negative one to the k plus three over

00:02:10.465 --> 00:02:14.211
the square root of k plus two.

00:02:14.235 --> 00:02:17.198
And it's just going to go
on and on and on like that.

00:02:17.732 --> 00:02:20.683
Now what we know already or
what we've seen an example of

00:02:20.683 --> 00:02:23.242
and we'll verify it or at
least conceptual verification

00:02:23.242 --> 00:02:27.872
for us at the end of this
video is the absolute

00:02:27.872 --> 00:02:32.872
value of this entire sum
is going to be less than

00:02:34.022 --> 00:02:36.397
the absolute value of the first term.

00:02:36.397 --> 00:02:37.842
So let me write that down.

00:02:37.842 --> 00:02:42.842
The absolute value of this
entire remainder R sub k...

00:02:42.881 --> 00:02:45.280
And some people refer
to this as kind of the

00:02:45.280 --> 00:02:48.232
alternating series remainder property

00:02:48.232 --> 00:02:49.998
or whatever you want to call it.

00:02:49.998 --> 00:02:52.184
But the absolute value
of this entire thing

00:02:52.184 --> 00:02:55.671
is going to be less than or
equal to the absolute value

00:02:55.671 --> 00:03:00.671
of the first term, negative
one to the k plus two

00:03:00.848 --> 00:03:04.822
over the square root of k plus one.

00:03:04.822 --> 00:03:08.631
And of course we want that to
be less than one thousandth.

00:03:08.631 --> 00:03:13.631
So that needs to be less
than or equal to 0.001.

00:03:14.786 --> 00:03:17.480
And so now this setup
right over here inspires

00:03:17.480 --> 00:03:19.752
you, I once again encourage
you to pause the video

00:03:19.752 --> 00:03:21.574
and see if you can figure out what is

00:03:21.574 --> 00:03:26.266
the minimum k that
satisfies this inequality.

00:03:27.422 --> 00:03:30.690
Well once again I'm assuming
you've had a go at it.

00:03:30.690 --> 00:03:32.820
And the key realization here is that this

00:03:32.820 --> 00:03:35.174
negative one to the k plus
two this makes this whole

00:03:35.174 --> 00:03:36.435
thing either positive or negative.

00:03:36.435 --> 00:03:38.514
The denominator is going to be positive.

00:03:38.514 --> 00:03:40.826
Well it's definitely going
to be positive for all

00:03:40.826 --> 00:03:43.089
the ks frankly for
which the principal root

00:03:43.089 --> 00:03:44.691
is going to be defined, but obviously

00:03:44.691 --> 00:03:47.320
these are all positive ks as well.

00:03:47.320 --> 00:03:50.128
So the numerator here just flips the sign.

00:03:50.128 --> 00:03:52.999
We flip between negative
one or positive one,

00:03:52.999 --> 00:03:54.616
negative one or positive one.

00:03:54.616 --> 00:03:58.074
But if we take the
absolute value whether it's

00:03:58.074 --> 00:04:00.358
negative or positive it's
going to end up being positive.

00:04:00.358 --> 00:04:05.358
So this thing on the left
hand side that's the same

00:04:06.480 --> 00:04:11.480
thing as just one over the
square root of k plus one

00:04:13.886 --> 00:04:18.886
and then that's going to be
less than or equal to 0.001.

00:04:21.132 --> 00:04:24.544
And now we just have to
solve this inequality.

00:04:24.544 --> 00:04:28.105
Find out which ks satisfy this inequality.

00:04:28.105 --> 00:04:30.310
And so let's see, we
can multiply both sides

00:04:30.310 --> 00:04:32.547
by the square root of k plus one.

00:04:32.547 --> 00:04:35.135
So square root of k plus one so we can

00:04:35.135 --> 00:04:36.904
get this out of the denominator.

00:04:36.904 --> 00:04:39.990
And let's actually multiple
both sides times 1,000

00:04:39.990 --> 00:04:42.023
because this is a thousandth
and so we'll end up

00:04:42.023 --> 00:04:43.317
with a one on the right-hand side.

00:04:43.317 --> 00:04:47.748
So times 1,000, times 1,000.

00:04:47.748 --> 00:04:50.595
And what we're going to be left with is

00:04:50.595 --> 00:04:53.165
these cancel out on the left-hand side.

00:04:53.672 --> 00:04:54.884
We don't have to flip the sign because

00:04:54.884 --> 00:04:56.789
we just multiplied by positive things.

00:04:56.789 --> 00:05:00.958
So we have 1,000 is less than or equal to

00:05:00.958 --> 00:05:03.629
the square root of k plus one.

00:05:03.629 --> 00:05:07.312
We could square both
sides and we get 1,000,000

00:05:09.065 --> 00:05:13.344
is less than or equal to k plus one.

00:05:13.344 --> 00:05:17.304
And then we can just
subtract one from both sides

00:05:17.304 --> 00:05:21.855
and we have 999,999 is going

00:05:21.855 --> 00:05:24.710
to be less than or equal to k.

00:05:24.710 --> 00:05:29.710
So k just has to be greater
than or equal to 999,999.

00:05:30.192 --> 00:05:32.157
And remember we want the smallest k

00:05:32.157 --> 00:05:33.539
that satisfies these conditions.

00:05:33.539 --> 00:05:34.556
So the smallest k if k has to be

00:05:34.556 --> 00:05:36.254
greater than or equal to this,

00:05:36.254 --> 00:05:41.254
the smallest k that satisfies
this is k is equal to 999,999.

00:05:42.252 --> 00:05:43.811
So let's write that down.

00:05:44.201 --> 00:05:46.671
So the smallest k for which this is true

00:05:46.671 --> 00:05:51.671
is going to be k is equal to 999,999.

00:05:53.541 --> 00:05:56.656
Now let's convince ourselves
that that remainder,

00:05:56.656 --> 00:05:58.538
that the absolute value
of the remainder is

00:05:58.538 --> 00:06:01.416
definitely going to be less...

00:06:04.573 --> 00:06:06.002
If we take the partial sum of the first

00:06:06.002 --> 00:06:09.910
999,999 terms that this remainder is

00:06:09.910 --> 00:06:12.766
actually going to be less than this.

00:06:12.766 --> 00:06:14.532
I'm telling you so far
we've just kind of worked

00:06:14.532 --> 00:06:15.685
from the premise that it will be,

00:06:15.685 --> 00:06:18.461
but let's actually look at
that and feel good about it.

00:06:18.461 --> 00:06:19.471
And once again encourage you to

00:06:19.471 --> 00:06:21.384
pause the video and try it on your own.

00:06:22.306 --> 00:06:23.765
So let's just rewrite this again,

00:06:23.765 --> 00:06:26.103
but I'm going to expand out R sub k.

00:06:26.103 --> 00:06:29.789
So we're saying that it's
all going to converge to S

00:06:29.789 --> 00:06:33.949
and we're going to take the
partial sum of the first

00:06:33.949 --> 00:06:38.569
999,999 terms and we're
claiming that this is going

00:06:38.569 --> 00:06:41.347
to be within one thousandth
of this right over here.

00:06:41.801 --> 00:06:43.685
So this is going to be plus and

00:06:43.685 --> 00:06:47.507
so the first term of the remainder,

00:06:47.507 --> 00:06:52.507
of R sub k is going to
be our millionth term.

00:06:54.299 --> 00:06:59.299
And so the millionth term, we
just have to remind ourselves,

00:06:59.417 --> 00:07:03.502
is going to be negative one
to the million and oneth power

00:07:03.502 --> 00:07:05.502
so negative one to the
million and oneth power

00:07:05.502 --> 00:07:08.306
that's going to be negative one

00:07:08.306 --> 00:07:09.615
because that's an odd number.

00:07:09.615 --> 00:07:11.249
So it's going to be negative one over

00:07:11.249 --> 00:07:13.012
the square root of a million.

00:07:13.012 --> 00:07:14.253
So actually let me do it like this.

00:07:14.253 --> 00:07:17.090
So it's going to be minus one

00:07:17.090 --> 00:07:21.326
over the square root of a million.

00:07:22.190 --> 00:07:24.161
And then the next term is going to be plus

00:07:24.161 --> 00:07:29.161
one over the square root
of a million and one.

00:07:30.667 --> 00:07:32.456
And it's positive now
because this is gonna be

00:07:33.779 --> 00:07:37.475
negative one to the
million and second power.

00:07:37.475 --> 00:07:41.791
And then it's going to be minus one over

00:07:41.791 --> 00:07:46.791
the square root of one million and two.

00:07:48.004 --> 00:07:51.274
And then plus, I'll
just do two more terms.

00:07:51.274 --> 00:07:56.274
One over the square root
of a million and three.

00:07:58.896 --> 00:08:03.896
And then minus one over the square root of

00:08:04.817 --> 00:08:08.132
a million and four and then plus, minus,

00:08:08.132 --> 00:08:12.074
it just keeps going on and
on, on and on and on forever.

00:08:12.074 --> 00:08:15.613
Now I want to convince
ourselves that this thing

00:08:15.613 --> 00:08:18.921
is going to be, the
absolute value of this thing

00:08:18.921 --> 00:08:21.483
is going to be less than one thousandth.

00:08:21.483 --> 00:08:22.761
It's going to be essentially less than the

00:08:22.761 --> 00:08:25.612
absolute value of this first term.

00:08:25.612 --> 00:08:26.835
So how do we think about it?

00:08:26.835 --> 00:08:28.047
So this first term we already know,

00:08:28.047 --> 00:08:32.625
this right over here is this
first term right over here is

00:08:32.625 --> 00:08:37.625
negative 0.001 because
it's one over one thousand

00:08:41.325 --> 00:08:43.685
and we have this negative right over here.

00:08:43.685 --> 00:08:45.556
Now the first thing we could realize,

00:08:45.556 --> 00:08:48.522
if we just put parentheses like this,

00:08:52.522 --> 00:08:54.534
we see that this is going to be positive,

00:08:54.534 --> 00:08:56.045
this term is larger than this term,

00:08:56.045 --> 00:08:57.133
this term is larger than this term,

00:08:57.133 --> 00:08:58.426
and we could keep going.

00:09:02.963 --> 00:09:06.163
So the remainder kind of starts
at negative one thousandth

00:09:06.163 --> 00:09:08.521
and then it just adds a
bunch of positive terms.

00:09:08.521 --> 00:09:11.474
So it's not going to get
any more negative than this.

00:09:11.474 --> 00:09:13.926
So we know that the remainder
can't get more negative

00:09:13.926 --> 00:09:16.342
than this, but let's
also verify that it can't

00:09:16.972 --> 00:09:20.352
get more than a positive
thousandth either.

00:09:20.352 --> 00:09:23.353
And the way that we can
do that is to just look

00:09:23.353 --> 00:09:25.326
at the parentheses in a different way.

00:09:25.326 --> 00:09:30.326
If we do it this way that's
going to be a negative value.

00:09:33.229 --> 00:09:35.691
This right over here is
going to be a negative value.

00:09:35.691 --> 00:09:40.060
And we can say plus that, plus that,

00:09:40.060 --> 00:09:40.989
and so we're going to have a

00:09:40.989 --> 00:09:42.475
bunch of negative values together.

00:09:42.475 --> 00:09:45.195
So just like that by really looking at the

00:09:45.195 --> 00:09:47.168
parentheses a little bit
differently we're able

00:09:47.168 --> 00:09:48.792
to say that this is definitely
going to be negative.

00:09:48.792 --> 00:09:51.224
This remainder is definitely
going to be a negative value,

00:09:51.224 --> 00:09:55.326
but it can't be any more
negative than our first term.

00:09:55.326 --> 00:09:57.767
So that tells us that the
absolute value of our remainder

00:10:03.166 --> 00:10:06.527
can't get any larger than one thousandth.

00:10:07.365 --> 00:10:09.364
As we add more and more
and more terms here

00:10:09.364 --> 00:10:12.280
we're not going to get
any further away from

00:10:13.172 --> 00:10:18.172
the actual sum than the first term.


Alternating series remainder

WEBVTT
Kind: captions
Language: en

00:00:00.766 --> 00:00:03.800
- [Voiceover] Let's explore
the infinite series.

00:00:04.433 --> 00:00:06.134
We're going to start at n equals one,

00:00:06.368 --> 00:00:10.600
and go to infinity of
negative one to the n plus one

00:00:10.966 --> 00:00:14.166
over n squared, which is
going to be equal to ...

00:00:14.166 --> 00:00:16.801
Let's see, when n is one,
this is going to be positive.

00:00:16.833 --> 00:00:18.167
It's going to be one.

00:00:18.167 --> 00:00:20.967
This, you go minus one over two squares,

00:00:20.967 --> 00:00:22.200
is minus 1/4

00:00:22.566 --> 00:00:24.200
plus 1/9

00:00:24.633 --> 00:00:28.333
minus 1/16

00:00:28.637 --> 00:00:30.434
plus 1/25 ...

00:00:30.434 --> 00:00:32.100
I'm actually going to go pretty far ...

00:00:32.100 --> 00:00:34.470
Minus 1/36,

00:00:34.833 --> 00:00:37.668
plus 1/49,

00:00:37.966 --> 00:00:40.269
minus 1/64.

00:00:41.101 --> 00:00:42.877
Yeah, that's pretty good.

00:00:42.877 --> 00:00:43.434
I'll stop there.

00:00:43.434 --> 00:00:45.633
Of course, we keep going on and on and on,

00:00:45.633 --> 00:00:48.105
and it's an alternating
series, plus, minus,

00:00:48.105 --> 00:00:50.666
just keeps going on and
on and on and on forever.

00:00:51.233 --> 00:00:53.366
Now, we know from previous tests,

00:00:53.366 --> 00:00:55.500
in fact, the alternating series test,

00:00:55.500 --> 00:00:57.104
that this satisfies the constraints

00:00:57.104 --> 00:00:59.766
of the alternating series test,

00:00:59.766 --> 00:01:01.466
and we're able to show that it converges.

00:01:01.900 --> 00:01:03.900
What we're doing now is,
actually trying to estimate

00:01:03.900 --> 00:01:05.533
what things converge to.

00:01:08.334 --> 00:01:11.466
We want to estimate
what this value, S, is.

00:01:11.500 --> 00:01:12.766
We're going to do that by doing

00:01:12.766 --> 00:01:15.233
a finite number of calculations,

00:01:15.233 --> 00:01:18.000
by not having to add this
entire thing together.

00:01:18.701 --> 00:01:20.433
Let's estimate it by taking, let's say,

00:01:20.433 --> 00:01:22.533
the partial sum of the first four terms.

00:01:23.234 --> 00:01:26.133
Let's take these four
terms right over here.

00:01:26.566 --> 00:01:29.933
Let's call that, that's
going to be S sub four.

00:01:30.336 --> 00:01:31.700
Then you're going to have a remainder,

00:01:31.700 --> 00:01:33.000
which is going to be everything else.

00:01:33.367 --> 00:01:34.570
All of this other stuff,

00:01:34.570 --> 00:01:36.133
I don't want even the brackets to end.

00:01:36.166 --> 00:01:38.133
That's going to be your remainder,

00:01:38.236 --> 00:01:42.000
the remainder, to get
to your actually sum,

00:01:42.300 --> 00:01:43.433
or whatever's left over

00:01:43.433 --> 00:01:45.300
when you just take the first four terms.

00:01:45.300 --> 00:01:48.269
This is from the fifth term
all the way to infinity.

00:01:48.634 --> 00:01:49.766
We've seen this before.

00:01:49.766 --> 00:01:52.466
The actual sum is going to
be equal to this partial sum

00:01:52.700 --> 00:01:54.466
plus this remainder.

00:01:54.966 --> 00:01:56.534
Well, we can calculate this.

00:01:56.700 --> 00:01:58.033
This is going to be, let's see ...

00:01:58.033 --> 00:01:59.933
Common denominator here,

00:02:00.034 --> 00:02:02.866
see, nine times 16 is 144.

00:02:02.900 --> 00:02:05.666
That's going to be 144,

00:02:05.966 --> 00:02:10.966
and then that's going
to be 144 minus 36/144,

00:02:11.771 --> 00:02:16.666
plus 16/144, minus 9/144.

00:02:16.700 --> 00:02:20.700
Let's see, that is 144,
negative 36 plus 16

00:02:20.700 --> 00:02:25.666
is minus 20, so it's
124 minus nine, is 115.

00:02:26.166 --> 00:02:28.100
This is all going to be

00:02:28.100 --> 00:02:31.966
equal to 115/144.

00:02:31.966 --> 00:02:33.469
I didn't even need a
calculator to figure that out.

00:02:33.700 --> 00:02:35.666
Plus some remainder.

00:02:36.366 --> 00:02:38.333
Plus some remainder.

00:02:39.500 --> 00:02:42.200
So, if we could figure out
some bounds on this remainder,

00:02:42.433 --> 00:02:47.433
we will figure out the
bounds on our actual sum.

00:02:48.900 --> 00:02:49.900
We'll be able to figure out,

00:02:49.900 --> 00:02:53.133
"Well, how far is this away
from this right over here?"

00:02:53.833 --> 00:02:55.833
There's two ways to think about it.

00:02:55.833 --> 00:02:57.733
Let's look at it.

00:02:57.900 --> 00:03:00.567
The first thing I want to see is,

00:03:00.567 --> 00:03:03.433
I want to show you that this
remainder right over here

00:03:03.433 --> 00:03:05.866
is definitely going to be positive.

00:03:06.233 --> 00:03:07.966
I actually encourage
you to pause the video

00:03:07.966 --> 00:03:09.966
and see if you can prove to yourself

00:03:09.966 --> 00:03:11.900
that this remainder over here

00:03:11.900 --> 00:03:15.600
is definitely going to be positive.

00:03:16.700 --> 00:03:18.766
I'm assuming you've had a go at it.

00:03:18.766 --> 00:03:20.066
Let's write the remainder down.

00:03:20.966 --> 00:03:22.300
Actually, I'll just write it ...

00:03:22.300 --> 00:03:24.066
Actually, I'll write it up here.

00:03:24.503 --> 00:03:29.503
R sub four is 1/25.

00:03:31.516 --> 00:03:32.966
Actually, I don't even have
to write it separately.

00:03:32.966 --> 00:03:34.900
I could show you in just right over here

00:03:34.900 --> 00:03:36.533
that this is going to be positive.

00:03:36.570 --> 00:03:37.700
How do I show that?

00:03:37.700 --> 00:03:39.066
Well, we just pair ...

00:03:39.100 --> 00:03:40.933
Let's just put some parentheses in here,

00:03:41.433 --> 00:03:43.066
and just pair these terms like this.

00:03:43.633 --> 00:03:46.000
1/25 minus 1/36.

00:03:46.033 --> 00:03:48.466
1/36th is less than 1/25.

00:03:48.833 --> 00:03:50.400
This one's positive, this one's negative.

00:03:50.833 --> 00:03:52.266
So this is positive.

00:03:52.566 --> 00:03:53.733
Then you have a positive term.

00:03:53.969 --> 00:03:56.666
Subtracting from that,
a smaller negative term.

00:03:56.700 --> 00:03:57.933
So this is going to be positive.

00:03:58.433 --> 00:04:01.233
So, if you just pair all these terms up,

00:04:01.233 --> 00:04:02.833
you're just going to have a whole series

00:04:02.833 --> 00:04:05.066
of positive terms.

00:04:05.434 --> 00:04:07.900
Just like that, we have established

00:04:07.900 --> 00:04:12.033
that R sub four, or R
four, we could call it,

00:04:12.033 --> 00:04:14.733
is going to be greater than zero.

00:04:15.700 --> 00:04:19.533
R four is going to be greater than zero.

00:04:20.500 --> 00:04:22.833
Now, the other thing I
want to prove is that

00:04:22.833 --> 00:04:24.166
this remainder is going to be

00:04:24.166 --> 00:04:28.200
less than the first term
that we haven't calculated,

00:04:28.500 --> 00:04:31.400
that the remainder is
going to be less than 1/25.

00:04:31.966 --> 00:04:34.100
Once again, I encourage
you to pause the video

00:04:34.100 --> 00:04:36.833
and see if you can put
some parentheses here

00:04:36.833 --> 00:04:38.501
in a certain way that will convince you

00:04:38.501 --> 00:04:41.100
that this entire infinite sum here,

00:04:41.100 --> 00:04:42.966
this remainder, is going to sum up

00:04:42.966 --> 00:04:45.666
to something that's less
than this first term.

00:04:47.166 --> 00:04:49.133
Once again, I'm assuming
you've had a go at it,

00:04:49.300 --> 00:04:50.400
so let's just write it down.

00:04:51.662 --> 00:04:53.000
I'll do that same pink color.

00:04:53.567 --> 00:04:56.433
Our remainder, when we
take the partial sum

00:04:56.433 --> 00:04:57.733
of the first four terms,

00:04:58.166 --> 00:04:59.533
it's 1/25.

00:04:59.833 --> 00:05:01.100
The way I'm going to write it,

00:05:01.100 --> 00:05:03.101
instead of writing minus 1/36,

00:05:03.101 --> 00:05:04.301
I'm going to write minus,

00:05:04.301 --> 00:05:05.700
I'm going to put the parentheses now

00:05:06.346 --> 00:05:08.000
around the second and third terms.

00:05:08.033 --> 00:05:12.866
This is going to be 1/36 minus 1/49.

00:05:13.766 --> 00:05:14.900
Then we're going to have

00:05:14.900 --> 00:05:19.900
minus 1/64 minus ...

00:05:20.033 --> 00:05:21.033
Actually, the next terms is going to be

00:05:21.033 --> 00:05:23.933
one over nine squared, 1/81.

00:05:24.566 --> 00:05:27.079
Then minus, and we keep going like that,

00:05:27.100 --> 00:05:30.200
on and on and on, on
and on and on, forever.

00:05:30.566 --> 00:05:31.866
Now, notice what happens.

00:05:32.300 --> 00:05:35.000
This, this term right over here

00:05:35.301 --> 00:05:36.600
is positive.

00:05:37.433 --> 00:05:39.900
We have a smaller number

00:05:39.900 --> 00:05:41.533
being subtracted from a larger number.

00:05:42.300 --> 00:05:44.466
This term right over here is positive.

00:05:45.100 --> 00:05:46.666
We're staring with 1/25,

00:05:46.966 --> 00:05:48.100
and then we're subtracting

00:05:48.100 --> 00:05:50.600
a bunch of positive things from it.

00:05:51.167 --> 00:05:55.400
This thing has to be less than 1/25.

00:05:56.367 --> 00:06:01.133
R sub four is going to be less than 1/25.

00:06:01.967 --> 00:06:04.700
Or, we could even write that as R sub four

00:06:04.700 --> 00:06:07.066
is less than 0.04.

00:06:07.300 --> 00:06:10.933
0.04, same things as 1/25.

00:06:11.500 --> 00:06:14.233
Actually, this logic right over here

00:06:14.233 --> 00:06:18.766
is the basis for the proof of
the alternating series test.

00:06:18.766 --> 00:06:20.402
This should make you feel pretty good,

00:06:20.402 --> 00:06:22.367
that, "Hey, look, this
thing is going to be

00:06:22.367 --> 00:06:23.969
"greater than zero,"

00:06:23.969 --> 00:06:27.600
and it's increasing, the more
terms that you add to it.

00:06:27.900 --> 00:06:30.867
But it's bounded from above.

00:06:30.900 --> 00:06:33.133
It's bounded from above at 1/25,

00:06:33.171 --> 00:06:34.367
which is a pretty good sense that hey,

00:06:34.367 --> 00:06:35.433
this thing is going to converge.

00:06:35.433 --> 00:06:36.633
But that's not what we're going to

00:06:36.633 --> 00:06:37.967
concern ourselves with here.

00:06:37.967 --> 00:06:39.600
Here, we just care about this range.

00:06:39.966 --> 00:06:42.333
The sum is the sum of these two things.

00:06:42.842 --> 00:06:46.700
So the entire sum is going to be

00:06:46.700 --> 00:06:51.700
less than 115/144

00:06:53.033 --> 00:06:55.535
plus the upper bound on R four.

00:06:56.100 --> 00:07:00.805
Plus 0.04, and it's
going to be greater than,

00:07:01.368 --> 00:07:04.333
it's going to be greater than,

00:07:05.300 --> 00:07:08.666
it's going to be greater than
our partial sum plus zero,

00:07:09.034 --> 00:07:11.200
because this remainder is
definitely greater than zero.

00:07:11.340 --> 00:07:12.110
You could just say,

00:07:12.110 --> 00:07:15.000
it's going to be greater
than our partial sum.

00:07:15.366 --> 00:07:16.233
And just like that,

00:07:16.233 --> 00:07:18.867
just doing a calculation that
I was able to do with hand,

00:07:19.033 --> 00:07:21.100
we're able to get pretty nice bounds

00:07:21.100 --> 00:07:23.000
around this infinite series.

00:07:23.233 --> 00:07:24.533
Infinite series.

00:07:25.171 --> 00:07:26.634
Let's now get the calculator out,

00:07:26.634 --> 00:07:28.533
just to get a little bit
better sense of things.

00:07:28.633 --> 00:07:32.933
If we say 115 divided by 144,

00:07:33.900 --> 00:07:37.200
that's .79861 repeating.

00:07:37.566 --> 00:07:42.566
This is 0.79861 repeating,

00:07:44.038 --> 00:07:46.066
is less than S,

00:07:46.300 --> 00:07:50.333
which is less than this thing plus .04.

00:07:50.900 --> 00:07:52.033
Let me write that down.

00:07:53.448 --> 00:07:58.448
Plus .04 gets us to .83861 repeating,

00:08:00.166 --> 00:08:02.333
83861 repeating.

00:08:02.900 --> 00:08:04.400
Actually, I could have
done that in my head.

00:08:04.502 --> 00:08:06.133
I don't know why I
resorted to a calculator.

00:08:07.500 --> 00:08:12.403
0.83861 repeating.

00:08:12.788 --> 00:08:13.433
And just like that,

00:08:13.433 --> 00:08:16.633
just a calculation we're
able to do by hand,

00:08:16.633 --> 00:08:17.700
we were able to come up

00:08:17.700 --> 00:08:21.366
with a pretty good approximation for S.

00:08:21.366 --> 00:08:23.900
And the big takeaway from here ...

00:08:23.900 --> 00:08:24.833
We're going to build on this,

00:08:24.833 --> 00:08:26.105
but this was really to
give you the intuition

00:08:26.105 --> 00:08:27.433
with a very concrete example,

00:08:27.433 --> 00:08:29.900
is when you have an
alternating series like this,

00:08:29.900 --> 00:08:32.300
the type of alternating
series that satisfies

00:08:32.300 --> 00:08:34.700
the alternating series test,

00:08:34.700 --> 00:08:36.700
where you can write it
as negative one to the n,

00:08:36.700 --> 00:08:38.166
or negative one to the n plus one,

00:08:38.166 --> 00:08:40.766
times a series of positive terms

00:08:40.766 --> 00:08:43.368
that are decreasing and
whose limits go to zeros

00:08:43.368 --> 00:08:44.666
and approaches infinity,

00:08:45.033 --> 00:08:46.833
not only do those things,

00:08:46.833 --> 00:08:48.666
not only do those things converge,

00:08:48.904 --> 00:08:51.303
but you can estimate your error

00:08:51.303 --> 00:08:54.700
based on the first term
that you're not including.

00:08:54.700 --> 00:08:56.566
Now, this was one example.

00:08:56.566 --> 00:08:58.100
It's going to be different depending on

00:08:58.100 --> 00:09:00.535
whether the first term
is negative or positive,

00:09:00.566 --> 00:09:01.433
and we're going to have to introduce

00:09:01.433 --> 00:09:03.533
the idea of absolute value
there, the magnitude.

00:09:03.706 --> 00:09:05.500
But the big takeaway here is

00:09:05.500 --> 00:09:08.767
that the magnitude of
your error is going to be

00:09:08.767 --> 00:09:13.767
no more than the magnitude
of the first term

00:09:14.100 --> 00:09:16.733
that you're not including
in your partial sum.


Ratio test

WEBVTT
Kind: captions
Language: en

00:00:00.624 --> 00:00:01.442
- [Voiceover] We already have a lot

00:00:01.442 --> 00:00:03.722
of experience with the geometric series.

00:00:03.956 --> 00:00:06.577
For example, if I have the
infinite geometric series

00:00:07.101 --> 00:00:11.406
starting at N equals K to
infinity of R to the N,

00:00:11.406 --> 00:00:16.242
which would be R to the K
plus R to the K plus one

00:00:16.242 --> 00:00:18.922
plus R to the K plus two and keep

00:00:18.922 --> 00:00:21.779
going on and on and on forever.

00:00:21.779 --> 00:00:23.991
There are a few things we've
already thought about here.

00:00:23.991 --> 00:00:26.240
We know what the common ratio is.

00:00:26.240 --> 00:00:31.240
So the common ratio,
which is the ratio between

00:00:31.558 --> 00:00:36.558
consecutive terms, is going
to be R to the K plus one.

00:00:36.956 --> 00:00:39.956
Actually, let me just write
this as R to the N plus one.

00:00:39.956 --> 00:00:42.473
I don't want to fixate on
this K right over here.

00:00:42.473 --> 00:00:46.273
So R to the N plus one over R to the N.

00:00:46.742 --> 00:00:48.985
Well, this is just going to be equal to R.

00:00:48.985 --> 00:00:51.374
Anything to the N plus one over
that same thing with the N,

00:00:51.374 --> 00:00:54.794
that's just going to be equal
to R, or R to the first power,

00:00:54.794 --> 00:00:56.004
and you see that here.

00:00:56.004 --> 00:00:57.573
When you go from one term to another

00:00:57.573 --> 00:01:00.974
you are just multiplying, you
are just multiplying by R,

00:01:01.491 --> 00:01:04.375
and this is all review.

00:01:04.375 --> 00:01:05.743
If it is not review, I encourage you

00:01:05.743 --> 00:01:08.440
to watch the videos on geometric series.

00:01:08.440 --> 00:01:10.656
And what's interesting
about this is we've proven

00:01:10.656 --> 00:01:13.125
to ourselves, in the videos
about geometric series,

00:01:16.617 --> 00:01:19.339
that if the common ratio,
if the absolute value

00:01:19.339 --> 00:01:22.408
of the common ratio is less than one,

00:01:24.608 --> 00:01:28.291
then the series converges.

00:01:29.625 --> 00:01:32.257
And if the absolute value of R

00:01:32.257 --> 00:01:34.575
is greater than or equal to one,

00:01:34.575 --> 00:01:35.988
then the series diverges.

00:01:35.988 --> 00:01:37.693
And that makes sense,
we've proven it as well,

00:01:37.693 --> 00:01:39.457
but it also makes
logical sense that, look,

00:01:39.457 --> 00:01:41.892
if that absolute value
of R is less than one

00:01:41.892 --> 00:01:45.493
then each term here is going
to go down by that common,

00:01:45.493 --> 00:01:47.659
or it's going to be multiplied
by that common ratio,

00:01:47.659 --> 00:01:50.659
and it's going to decrease
on and on and on and on

00:01:50.659 --> 00:01:52.630
until it makes sense
that, even though this

00:01:52.630 --> 00:01:56.459
is an infinite sum, it will
converge to a finite value.

00:01:57.060 --> 00:01:59.459
Now, with that out of the way for review,

00:01:59.459 --> 00:02:04.459
let's tackle something a
little bit more interesting.

00:02:05.077 --> 00:02:07.493
Let's say that we want to look at a,

00:02:07.493 --> 00:02:11.526
we want to figure out
whether a series like this,

00:02:11.671 --> 00:02:13.793
so starting at N equals five to infinity

00:02:13.793 --> 00:02:17.180
of, let's say, N to the tenth power.

00:02:17.180 --> 00:02:19.577
The numerator is growing quickly.

00:02:19.577 --> 00:02:23.492
N to the tenth power over N factorial,

00:02:23.492 --> 00:02:25.876
and factorial we know also
grows very, very quickly,

00:02:25.876 --> 00:02:27.127
probably it grows much faster than

00:02:27.127 --> 00:02:31.460
even a high degree polynomial like this,

00:02:31.460 --> 00:02:33.243
or a high degree term like this,

00:02:33.243 --> 00:02:36.527
but how do we prove that it converges?

00:02:36.527 --> 00:02:38.293
We could definitely the diverges test

00:02:38.293 --> 00:02:41.065
to show that this does not diverge,

00:02:41.065 --> 00:02:44.078
but how do we prove that
this actually converges?

00:02:44.078 --> 00:02:47.543
And maybe we can use a little
bit of our intuition here.

00:02:47.959 --> 00:02:50.627
Well, let's see if we can
come up with a common ratio.

00:02:50.627 --> 00:02:53.144
So let's see if there's
a common ratio here.

00:02:53.144 --> 00:02:55.460
So let's take the N plus oneth term,

00:02:55.460 --> 00:02:59.841
which is going to be N
plus one to the tenth power

00:02:59.841 --> 00:03:04.841
over N plus one factorial and
divide that by the Nth term,

00:03:06.392 --> 00:03:11.392
so let's divide that by N to
the tenth over N factorial.

00:03:12.826 --> 00:03:15.243
Well, dividing by a fraction,
or dividing by anything,

00:03:15.243 --> 00:03:17.243
is equivalent to multiplying
by it's reciprocal,

00:03:17.243 --> 00:03:19.309
So let's just multiply by the reciprocal,

00:03:19.309 --> 00:03:24.001
so times N factorial over N to the tenth.

00:03:24.001 --> 00:03:26.592
Remember all I'm trying to do
is exactly right over here.

00:03:26.592 --> 00:03:30.043
See if there is some
type of a common ratio.

00:03:30.812 --> 00:03:32.259
Well, if we do a little algebra here

00:03:32.259 --> 00:03:36.341
N plus one factorial, and
factorial algebra is always fun.

00:03:36.341 --> 00:03:41.341
This is the same thing as N
plus one times N factorial.

00:03:43.043 --> 00:03:45.513
Times N factorial.

00:03:45.513 --> 00:03:46.698
You're not use to seeing order

00:03:46.698 --> 00:03:48.260
of operations with factorials,

00:03:48.260 --> 00:03:51.359
but this factorial only applies
to this N right over here,

00:03:51.359 --> 00:03:52.877
and why is that useful?

00:03:52.877 --> 00:03:55.626
Because this N factorial
cancels with this N factorial,

00:03:55.626 --> 00:04:00.626
and we're left with N plus
one to the tenth power

00:04:00.758 --> 00:04:05.758
over N plus one times N to the tenth.

00:04:08.376 --> 00:04:10.505
Times N to the tenth.

00:04:10.505 --> 00:04:11.691
So I know what you're thinking.

00:04:11.691 --> 00:04:14.145
"Hey, wait, look, this isn't
a fixed common ratio here."

00:04:14.145 --> 00:04:16.545
The ratio between consecutive terms here,

00:04:16.545 --> 00:04:19.361
when I took the N plus oneth
term divided by the Nth term,

00:04:19.361 --> 00:04:22.009
it's changing depending on what my N is.

00:04:22.009 --> 00:04:25.996
This ratio, I guess you
could say, is a function of N

00:04:25.996 --> 00:04:27.896
so this doesn't seem too useful,

00:04:27.896 --> 00:04:29.079
but what if I were to say,

00:04:29.079 --> 00:04:31.046
"O.K., well look, with any of these series

00:04:31.046 --> 00:04:33.546
"we really care about
the behavior as our N's

00:04:33.546 --> 00:04:35.777
"get really, really, really large as

00:04:35.777 --> 00:04:38.947
the limit, as our N's, go to infinity."

00:04:38.947 --> 00:04:40.878
So what if we were to
look at the behavior here,

00:04:40.878 --> 00:04:42.129
and if this is approaching some

00:04:42.129 --> 00:04:43.928
actual values as N approaches infinity,

00:04:43.928 --> 00:04:45.679
well, it would make conceptual sense

00:04:45.679 --> 00:04:47.212
that we could kind of think of that as

00:04:50.596 --> 00:04:52.129
the limit of our common ratio.

00:04:52.796 --> 00:04:54.177
So let's do that.

00:04:54.177 --> 00:04:59.177
Let's take the limit as N
approaches infinity of this thing.

00:05:00.329 --> 00:05:01.547
So remember what we're doing here.

00:05:01.547 --> 00:05:03.379
This isn't just kind of some voodoo.

00:05:03.379 --> 00:05:06.713
Conceptually, let me copy and paste this.

00:05:07.662 --> 00:05:09.262
Conceptually, we're just trying to think

00:05:09.262 --> 00:05:12.278
about what is the common ratio approach?

00:05:12.278 --> 00:05:14.212
What is the ratio between
consecutive terms?

00:05:14.212 --> 00:05:15.963
The N plus one term and the Nth term

00:05:15.963 --> 00:05:18.947
as N gets really, really,
really, really large?

00:05:19.347 --> 00:05:22.178
And what we see here is
we have this up here,

00:05:22.178 --> 00:05:23.213
we don't even have to multiply it out,

00:05:23.213 --> 00:05:24.743
this is going to be N to the tenth

00:05:24.743 --> 00:05:26.329
plus a bunch of other stuff.

00:05:26.329 --> 00:05:28.612
It's going to be a
tenth degree polynomial,

00:05:28.612 --> 00:05:31.070
and this down here is going to be,

00:05:31.070 --> 00:05:32.629
actually you can figure it out,

00:05:32.629 --> 00:05:36.064
this is going to be N to the
11th plus N to the tenth power,

00:05:36.972 --> 00:05:38.573
so the limit as N approaches infinity,

00:05:38.573 --> 00:05:40.287
well this denominator is going
to grow faster than that.

00:05:40.287 --> 00:05:41.504
You could divide the numerator

00:05:41.504 --> 00:05:43.156
and the denominator by N to the tenth,

00:05:46.662 --> 00:05:48.705
actually the numerator and the
denominator by N to the 11th,

00:05:48.705 --> 00:05:50.740
and everything up here
is going to go to zero,

00:05:50.740 --> 00:05:52.622
and this is going to be one,

00:05:52.622 --> 00:05:54.504
and so the limit, right over here,

00:05:57.148 --> 00:06:00.479
this limit is going to be equal to...

00:06:01.355 --> 00:06:04.522
This limit is going to be equal to zero.

00:06:04.522 --> 00:06:06.155
As N approaches infinity one way the ratio

00:06:06.155 --> 00:06:11.155
between consecutive terms approaches zero.

00:06:12.819 --> 00:06:17.772
So this seems, if we use
the logic from our common

00:06:17.772 --> 00:06:19.823
ratio up here when we
looked at geometric series,

00:06:19.823 --> 00:06:21.538
this is clearly not a geometric series,

00:06:21.538 --> 00:06:23.487
but we would say, "Well, look, as N gets

00:06:23.487 --> 00:06:25.439
really, really, really
large the ratio between

00:06:25.439 --> 00:06:27.638
"consecutive terms gets smaller
and smaller and smaller.

00:06:27.638 --> 00:06:30.689
"Hey, maybe we could
make the same conclusion.

00:06:31.105 --> 00:06:32.438
"Maybe we could make
the same conclusion that

00:06:32.438 --> 00:06:37.438
"therefore this series
actually converges."

00:06:38.073 --> 00:06:43.073
The series actually converges,
so N factorial converges.

00:06:43.465 --> 00:06:45.600
So can we make this statement?

00:06:45.600 --> 00:06:48.607
And the answer is yes we can,

00:06:48.607 --> 00:06:52.500
and what allows us to
do it is the ratio test.

00:06:52.531 --> 00:06:54.260
Is the ratio test.

00:06:54.260 --> 00:06:55.325
Let me write that down.

00:06:55.325 --> 00:06:57.501
The ratio test.

00:06:58.040 --> 00:07:00.873
And all the ratio test tells us is

00:07:00.873 --> 00:07:03.739
if we have an infinite series...

00:07:03.739 --> 00:07:04.969
If we have an infinite series,

00:07:04.969 --> 00:07:09.076
and I can say from N equals K to infinity,

00:07:09.076 --> 00:07:14.076
and if we take the limit
as N approaches infinity,

00:07:17.883 --> 00:07:22.324
of the absolute value
of A to the N plus one

00:07:22.324 --> 00:07:25.140
over A to the N, and over here I didn't

00:07:25.140 --> 00:07:26.660
take the absolute value of that limit,

00:07:26.660 --> 00:07:28.042
but we absolutely could have taken

00:07:28.042 --> 00:07:29.825
the absolute value of that limit

00:07:29.825 --> 00:07:33.075
if all of these terms right
over here were positive,

00:07:33.075 --> 00:07:35.026
so the absolute value would
be the same thing as itself.

00:07:36.118 --> 00:07:38.383
If this approaches some limit,

00:07:38.921 --> 00:07:40.443
if this approaches some limit,

00:07:40.443 --> 00:07:44.040
so once again, this limit is
what is going to be our ratio

00:07:44.040 --> 00:07:47.698
between consecutive terms
as N gets larger and larger,

00:07:48.467 --> 00:07:53.467
the ratio tells us if L is less than one,

00:07:55.342 --> 00:07:58.358
which was the situation here,
L is clearly less than one,

00:07:58.358 --> 00:08:00.509
then the series converges.

00:08:04.653 --> 00:08:07.908
Some definitions will say
it's absolutely convergent,

00:08:07.908 --> 00:08:09.826
it converges absolutely,
the absolute value

00:08:09.826 --> 00:08:12.258
of the series converges,
but then that also means

00:08:12.258 --> 00:08:14.246
that the series itself converges.

00:08:14.246 --> 00:08:19.246
If L is greater than one
then series diverges,

00:08:20.735 --> 00:08:22.903
Series diverges.

00:08:23.595 --> 00:08:28.192
and if L is equal to one it's
inconclusive, we don't know.

00:08:29.191 --> 00:08:31.091
We would have to do some other test

00:08:31.091 --> 00:08:34.659
to prove whether it converges or diverges.

00:08:35.193 --> 00:08:37.291
So that's the essence of the ratio test.

00:08:37.291 --> 00:08:39.758
Find the absolute value of the ratio

00:08:39.758 --> 00:08:41.408
between consecutive terms,

00:08:41.408 --> 00:08:43.127
take the limit as N approaches infinity,

00:08:43.127 --> 00:08:46.742
if that approaches an actual limit,

00:08:46.742 --> 00:08:51.742
and that limit is less than
one, then the series converges,

00:08:51.923 --> 00:08:53.798
and it's really based on
the same fundamental idea

00:08:53.798 --> 00:08:57.381
that we saw with the common
ratio of geometric series.


Worked example: Series estimation with integrals

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:02.512
- [Voiceover] In the last video we saw

00:00:02.512 --> 00:00:04.151
that if we have an infinite series

00:00:04.151 --> 00:00:06.052
where each term is a function of N,

00:00:06.052 --> 00:00:08.816
and the function itself
is a continuous, positive,

00:00:08.816 --> 00:00:13.299
decreasing function over
the intervals we care about,

00:00:13.299 --> 00:00:15.817
and we assume that this series converges,

00:00:15.817 --> 00:00:19.359
then we can estimate that series,

00:00:19.359 --> 00:00:21.385
we can estimate the value
that it converges to,

00:00:21.385 --> 00:00:26.250
using a finite number of
terms and some integrals,

00:00:26.250 --> 00:00:27.685
and the way that we
establish these integrals

00:00:27.685 --> 00:00:31.401
is because when we split
up the sum into finite sum

00:00:31.401 --> 00:00:32.886
and then another infinite series,

00:00:32.886 --> 00:00:34.517
we're about to conceptualize that

00:00:34.517 --> 00:00:36.435
infinite series in two different ways.

00:00:36.435 --> 00:00:38.996
It could be an underestimate of

00:00:39.380 --> 00:00:41.267
this integral right over here,

00:00:41.267 --> 00:00:43.250
or it could be an overestimate of

00:00:43.250 --> 00:00:44.768
this integral right over here,

00:00:44.768 --> 00:00:46.668
and because of that it
allows us to not only

00:00:46.668 --> 00:00:49.518
estimate S but to put bounds on it.

00:00:49.518 --> 00:00:51.134
If we can evaluate the left hand side here

00:00:51.134 --> 00:00:53.851
and the right hand side here
for a given infinite series,

00:00:53.851 --> 00:00:56.317
then we can establish our bounds on S,

00:00:56.317 --> 00:00:58.565
so let's see how good these are by

00:00:58.565 --> 00:01:01.985
applying it to a
particular infinite series.

00:01:02.367 --> 00:01:05.018
So let's say that the infinite series

00:01:05.018 --> 00:01:08.851
we want to apply it to, let
me do it in the yellow color,

00:01:09.016 --> 00:01:11.967
so it's going to be, let's say,

00:01:11.967 --> 00:01:15.851
we go from N equals one to
infinity of one over N squared,

00:01:15.851 --> 00:01:16.934
and let's say we didn't know how

00:01:16.934 --> 00:01:17.884
to find the exact value here,

00:01:17.884 --> 00:01:18.950
but we want to estimate it.

00:01:18.950 --> 00:01:23.950
Let's say we want to estimate
it using the first five terms,

00:01:25.635 --> 00:01:29.918
so we'll do S sub five is equal to the sum

00:01:30.994 --> 00:01:35.994
of N equals one to infinity,

00:01:37.463 --> 00:01:41.895
sorry, to five of one over n squared,

00:01:41.895 --> 00:01:45.924
and let's see, this is
going to be equal to,

00:01:45.924 --> 00:01:48.617
let's get a calculator
out, this is going to be

00:01:48.617 --> 00:01:49.902
one over one squared, which is just one,

00:01:49.902 --> 00:01:52.031
plus one over two squared,
which is one fourth,

00:01:52.031 --> 00:01:55.916
plus one over three
squared, which is one ninth,

00:01:55.916 --> 00:01:59.054
plus one over four
squared, which is one 16th,

00:01:59.823 --> 00:02:03.919
plus one over five squared,
which is going to be one 25th,

00:02:06.300 --> 00:02:09.884
and we get this value right over here,

00:02:09.884 --> 00:02:12.285
so I'll just say this is, I'll say it's

00:02:12.285 --> 00:02:14.650
approximately, one point four six four.

00:02:15.217 --> 00:02:18.469
Approximately one point four six four,

00:02:18.718 --> 00:02:21.617
and then if we evaluate
each of these integrals,

00:02:21.617 --> 00:02:25.584
so our K in this case is going to be five.

00:02:25.584 --> 00:02:26.751
I just picked that arbitrarily.

00:02:26.751 --> 00:02:28.701
We could probably get a better,
or we would get a better

00:02:28.701 --> 00:02:31.619
estimate if we had a higher K,
if we did the first 10 terms

00:02:31.619 --> 00:02:34.118
or 20 terms, and we would
have a worse estimate

00:02:34.118 --> 00:02:36.369
if we did K equals three, but
I just picked K equals five

00:02:36.369 --> 00:02:38.560
because it seems reasonable,
something that I could compute.

00:02:38.560 --> 00:02:40.001
I actually didn't need
a calculator to do that.

00:02:40.001 --> 00:02:41.084
I could have done it by hand.

00:02:41.084 --> 00:02:42.368
It would have just taken more time.

00:02:42.368 --> 00:02:44.342
Let's evaluate these integrals here.

00:02:44.788 --> 00:02:47.834
So the integral from five plus one,

00:02:47.834 --> 00:02:52.834
so it's six to infinity
of one over N squared DX,

00:02:54.623 --> 00:02:58.447
and, let's see, that's
going to be equal to,

00:02:58.447 --> 00:03:01.947
let's write it this way, this
is going to be the limit,

00:03:01.947 --> 00:03:04.398
I'll do all of it in blue actually.

00:03:04.398 --> 00:03:08.679
This is going to be equal to the limit as,

00:03:10.863 --> 00:03:13.781
let's introduce a new
variable, B approaches infinity

00:03:14.966 --> 00:03:17.029
of the integral from six to

00:03:17.029 --> 00:03:20.634
infinity of one over N squared DX,

00:03:20.634 --> 00:03:25.634
which is equal to the limit
as B approaches infinity of,

00:03:29.242 --> 00:03:30.535
let's see, the antiderivative of this

00:03:30.535 --> 00:03:35.103
is negative N to the negative one,

00:03:35.569 --> 00:03:38.686
and we're going to evaluate
that at B and at six,

00:03:38.686 --> 00:03:43.035
so this is equal to the
limit as B approaches

00:03:43.035 --> 00:03:46.086
infinity of negative one over B,

00:03:46.086 --> 00:03:49.352
negative one over B minus negative one

00:03:49.352 --> 00:03:52.734
over six, so plus one over six.

00:03:53.395 --> 00:03:54.237
Well as B approaches infinity

00:03:54.237 --> 00:03:56.103
this term right over here goes to zero,

00:03:56.103 --> 00:03:58.870
so this is just going to
be equal to one sixth,

00:03:59.236 --> 00:04:01.187
and we can use that exact same logic

00:04:01.187 --> 00:04:04.486
to evaluate this integral,
so let's do that.

00:04:05.084 --> 00:04:07.088
Actually, I'll do it right over here.

00:04:07.088 --> 00:04:10.019
So the improper integral,
I should say, from,

00:04:10.019 --> 00:04:11.302
and here we're going to evaluate not

00:04:11.302 --> 00:04:13.237
from six but from five, that's our K.

00:04:13.237 --> 00:04:17.736
Five to infinity of one over N squared DX.

00:04:17.736 --> 00:04:19.604
Well, the only difference here between

00:04:19.604 --> 00:04:21.968
these two integrals is this bottom bound.

00:04:21.968 --> 00:04:23.104
This was one over six,

00:04:23.104 --> 00:04:24.104
and by the same arugemnts,

00:04:24.104 --> 00:04:29.104
this one is going to end
up being one over five.

00:04:30.964 --> 00:04:32.354
And so now we can put everything

00:04:32.354 --> 00:04:34.214
into this compound inequality.

00:04:34.214 --> 00:04:37.701
We're going to have one
point four six four.

00:04:37.701 --> 00:04:42.432
One point one six four plus,
and I'm being a little bit

00:04:42.432 --> 00:04:44.764
approximate here, it's
really this S sub K,

00:04:44.764 --> 00:04:47.081
which we have the exact
value for in our calculator,

00:04:47.081 --> 00:04:51.181
so it's one point four
six four plus one sixth,

00:04:51.873 --> 00:04:56.314
plus one sixth, is less than or equal to,

00:04:57.098 --> 00:05:00.347
less than or equal to,
is less than or equal to

00:05:00.347 --> 00:05:05.347
our sum, which we care about,
which is less than or equal to

00:05:05.465 --> 00:05:10.465
one point four six four plus
one fifth, plus one fifth,

00:05:14.833 --> 00:05:17.264
and so this one is fairly
straight forward to calculate.

00:05:17.264 --> 00:05:19.082
This is going to be plus point two,

00:05:19.082 --> 00:05:20.946
so this is going to be less than

00:05:20.946 --> 00:05:25.946
or equal to one point two six four,

00:05:27.164 --> 00:05:30.082
so sum is going to be less
than or equal to that,

00:05:30.082 --> 00:05:34.847
and on this side if we took
our original expression,

00:05:34.847 --> 00:05:38.762
and then I added one sixth, one sixth,

00:05:38.762 --> 00:05:43.132
I get one point...I get
one...let's see, what did I?

00:05:43.132 --> 00:05:44.449
Oh, this is supposed to be one...

00:05:44.449 --> 00:05:45.831
I wrote this one wrong.

00:05:45.831 --> 00:05:49.383
O.K., so this is going to
be one point sixth three oh.

00:05:49.597 --> 00:05:50.782
Let me just write one point...

00:05:50.782 --> 00:05:55.315
So it's going to be
one point six three oh,

00:05:55.315 --> 00:05:58.848
and on this side, my
brain was malfunctioning.

00:05:59.346 --> 00:06:02.024
This should be one point one,

00:06:02.024 --> 00:06:03.581
I took away two instead of adding two,

00:06:03.581 --> 00:06:08.581
one point six six four,
one point six six four,

00:06:09.365 --> 00:06:10.416
and so just like that you see,

00:06:10.416 --> 00:06:15.416
O.K., this thing must be like
one point six three something.

00:06:17.170 --> 00:06:18.606
It could be one point four something.

00:06:18.606 --> 00:06:20.022
It could be one point five something.

00:06:20.022 --> 00:06:22.250
It could be one point six six,

00:06:22.250 --> 00:06:24.183
but this is already giving
us a lot of precision,

00:06:24.183 --> 00:06:26.184
if we only cared about
things to the 10ths place,

00:06:26.184 --> 00:06:27.702
this kind of, we're done.

00:06:27.702 --> 00:06:29.800
We know it's going to be
one point six something,

00:06:29.800 --> 00:06:31.884
and if we go a little bit further

00:06:31.884 --> 00:06:33.401
this is already giving
us a lot of precision,

00:06:33.401 --> 00:06:35.534
and you can imagine if
you added a few more terms

00:06:35.534 --> 00:06:39.066
in the partial sum, you would
get even more precision.

00:06:39.066 --> 00:06:41.485
So hopefully you found this
to be kind of neat exercise.

00:06:41.485 --> 00:06:43.466
Not only did we get a
good bit of precision,

00:06:43.466 --> 00:06:47.134
a good bit of precision with
just taking a partial sum

00:06:47.134 --> 00:06:48.617
and the help of these integrals,

00:06:48.617 --> 00:06:50.161
but we got a good bound on it.

00:06:50.161 --> 00:06:52.267
We know that the sum
that we're converging to

00:06:52.267 --> 00:06:55.322
is between these two values.


Series estimation with integrals

WEBVTT
Kind: captions
Language: en

00:00:00.442 --> 00:00:03.490
- [Voiceover] So let's say S
is the value that this infinite

00:00:03.490 --> 00:00:04.683
series converges to.

00:00:04.683 --> 00:00:08.350
We're going to assume that
this series actually converges.

00:00:08.635 --> 00:00:11.945
And the definition of the
series, each term is going

00:00:11.945 --> 00:00:13.258
to be a function of N.

00:00:13.258 --> 00:00:15.018
We're going to assume
that this is the same type

00:00:15.018 --> 00:00:16.618
of series that we looked
at when we looked at the

00:00:16.618 --> 00:00:19.810
integral test, or namely that
this function is a continuous

00:00:19.810 --> 00:00:23.226
positive decreasing
function over the interval

00:00:23.226 --> 00:00:24.669
that we care about.

00:00:24.818 --> 00:00:29.818
So it is continuous, it is
positive, and it is decreasing.

00:00:29.988 --> 00:00:31.748
This isn't saying
continuous plus decreasing.

00:00:31.748 --> 00:00:33.012
Let me just write it that way.

00:00:33.012 --> 00:00:38.012
Continuous positive and it is decreasing.

00:00:38.889 --> 00:00:41.837
So it could be a function that
looks something like this.

00:00:42.010 --> 00:00:45.488
Now, my goal of this video
is to see if we can estimate

00:00:45.488 --> 00:00:47.224
a range around S.

00:00:47.224 --> 00:00:48.904
So this is going to be very
useful, because we've seen

00:00:48.904 --> 00:00:51.481
some infinite series where
we are able to figure out

00:00:51.481 --> 00:00:54.224
what exactly does it converge to.

00:00:54.224 --> 00:00:56.143
But you could imagine
there are many, many more

00:00:56.143 --> 00:00:57.586
where we're not going
to be able to figure out

00:00:57.586 --> 00:00:59.554
exactly what it converges to.

00:00:59.554 --> 00:01:00.994
And instead we're going
to have to do it using a

00:01:00.994 --> 00:01:02.507
computer or by hand.

00:01:02.507 --> 00:01:05.266
And in those cases, it's
good to know how good

00:01:05.266 --> 00:01:06.650
our estimate is.

00:01:06.650 --> 00:01:08.866
And we also want as good
of an estimate as possible

00:01:08.866 --> 00:01:11.922
with as little computation as possible.

00:01:11.922 --> 00:01:13.621
So let's think about how we can do that.

00:01:13.826 --> 00:01:15.731
Well, the way to tackle it,
you could imagine, is let's

00:01:15.731 --> 00:01:18.753
split this up, this infinite
sum, let's split it up

00:01:18.753 --> 00:01:21.017
into the sum of a finite sum.

00:01:21.017 --> 00:01:23.341
So let's say the first k terms.

00:01:23.441 --> 00:01:26.805
So n equals one to k of f of n.

00:01:26.848 --> 00:01:28.418
So this is very computable.

00:01:28.418 --> 00:01:31.777
If k is low enough and if
f a simple enough function,

00:01:31.777 --> 00:01:33.252
you could probably do this by hand.

00:01:33.252 --> 00:01:35.645
But you could definitely
do this with a computer.

00:01:35.704 --> 00:01:39.159
And then it's going to
be that plus an infinite.

00:01:39.159 --> 00:01:41.343
Plus another infinite series.

00:01:41.343 --> 00:01:43.199
But now you're going to
pick up at k plus one, the

00:01:43.199 --> 00:01:44.654
k plus one term.

00:01:45.361 --> 00:01:48.587
And you're going to go
to infinity of f of n.

00:01:48.783 --> 00:01:51.639
So if we could put some
bounds on this, then that'll

00:01:51.639 --> 00:01:56.639
allow us to put some bounds
on this right over here.

00:01:56.677 --> 00:02:00.312
Because this is just the
sum of the partial sum of

00:02:00.312 --> 00:02:03.862
the partial sum of the first
k terms and the remainder

00:02:03.862 --> 00:02:08.781
that we get after to get
us to the actual value.

00:02:08.781 --> 00:02:11.845
So you could see kind of
what's left over after

00:02:11.845 --> 00:02:14.309
we take that partial sum.

00:02:14.309 --> 00:02:16.253
And this is easier to
write for me than this

00:02:16.253 --> 00:02:17.453
right over here.

00:02:17.453 --> 00:02:22.133
So they key is, can we come
up with some bounds for this?

00:02:23.148 --> 00:02:25.515
And to do that I'm going
to go to this graph and use

00:02:25.515 --> 00:02:27.642
some of the same arguments
we used, or the same

00:02:27.642 --> 00:02:30.679
conceptual ideas we used,
for the integration test.

00:02:32.041 --> 00:02:35.900
There's two ways to
conceptualize what the sum

00:02:35.900 --> 00:02:37.835
represents relative to this graph.

00:02:37.835 --> 00:02:41.657
As we'll see, it can
represent an over estimate of

00:02:41.657 --> 00:02:44.827
the area between sum,
x value, and infinity.

00:02:44.827 --> 00:02:46.973
And it could represent
the underestimate of a

00:02:46.973 --> 00:02:48.246
different region.

00:02:48.246 --> 00:02:49.237
So let's look at that.

00:02:49.237 --> 00:02:53.004
So let's first think
about the underestimate.

00:02:53.085 --> 00:02:54.982
Let's think about the underestimate.

00:02:54.982 --> 00:02:57.279
So if this right over here,
let's say that this right

00:02:57.279 --> 00:02:59.022
over here is k.

00:03:00.037 --> 00:03:02.209
Actually, let me do it
in a color that's...

00:03:02.373 --> 00:03:04.816
Let me do it in that yellow color.

00:03:05.093 --> 00:03:07.781
So let's say that this
right over here is k.

00:03:07.781 --> 00:03:09.661
This is k plus one.

00:03:13.168 --> 00:03:15.379
Let's do this is k plus two.

00:03:15.379 --> 00:03:19.467
K plus 2, k plus 3, on and on and on.

00:03:19.467 --> 00:03:23.548
So one way to conceptualize
this sum right over here

00:03:23.548 --> 00:03:26.948
is it could be the sum of
the following rectangles.

00:03:27.978 --> 00:03:31.331
So the first term is this
area, the area of this

00:03:31.331 --> 00:03:32.601
first rectangle.

00:03:32.601 --> 00:03:36.049
Because this area's height,
or this rectangle's height,

00:03:36.049 --> 00:03:38.041
is f of k plus one.

00:03:41.333 --> 00:03:44.033
And its width is one, so
f of k plus one times one.

00:03:44.033 --> 00:03:45.809
Its area is just going to
be f of k plus one, which

00:03:45.809 --> 00:03:47.960
is exactly this first term
right over here, where

00:03:47.960 --> 00:03:49.661
n is k plus one.

00:03:49.713 --> 00:03:51.784
And then the second term,
by that same argument,

00:03:51.784 --> 00:03:55.352
could represent the
area of this rectangle.

00:03:55.352 --> 00:03:57.703
The third term could represent
the area of the rectangle.

00:03:57.703 --> 00:04:00.423
And we could just keep
going on and on and on.

00:04:00.423 --> 00:04:04.087
So what are the sums of these
areas of these rectangles?

00:04:04.087 --> 00:04:06.068
What are the sums of
these terms representing?

00:04:06.068 --> 00:04:08.204
Well, you could view it as an estimate.

00:04:08.204 --> 00:04:11.705
You could view it as an
estimate of the area under

00:04:11.705 --> 00:04:15.638
the curve between x equals
k and x equals infinity.

00:04:15.638 --> 00:04:18.173
But it's going to be an underestimate.

00:04:18.173 --> 00:04:21.183
Notice these are all completely
contained in that area.

00:04:21.183 --> 00:04:25.006
So one way to think about
it is that our R sub k

00:04:27.559 --> 00:04:30.756
is less than or equal to its
underestimate for the area

00:04:30.756 --> 00:04:35.756
between x equals k and
infinity of f of x dx.

00:04:39.982 --> 00:04:44.085
So that essentially puts
an upper bound on us.

00:04:44.085 --> 00:04:46.287
And this is already
interesting, because now we can

00:04:46.287 --> 00:04:51.287
already say if S -- so we
know that S is equal to this.

00:04:52.718 --> 00:04:55.415
Now if this is less than
this thing right over here,

00:04:55.415 --> 00:04:58.503
we can say that S is going
to be less than or equal to

00:04:58.503 --> 00:05:03.503
our partial sum plus this thing.

00:05:05.480 --> 00:05:10.480
Plus the improper integral from
k to infinity of f of x dx.

00:05:11.462 --> 00:05:14.477
Notice if this is equal to
this and now since this is less

00:05:14.477 --> 00:05:16.781
than this, this must be
less than what we have

00:05:16.781 --> 00:05:18.078
on the right hand side.

00:05:18.078 --> 00:05:20.477
So just like that, if we're
able to compute these two things

00:05:20.477 --> 00:05:22.502
and we are often able to
compute these two things,

00:05:22.502 --> 00:05:26.195
we're able to put an upper
bound on our actual sum.

00:05:26.703 --> 00:05:28.736
Now what about placing
a lower bound on it?

00:05:29.366 --> 00:05:32.626
Well, we could conceptualize
the same sum, the same

00:05:32.626 --> 00:05:33.734
R sub k.

00:05:33.734 --> 00:05:38.734
Instead we can conceptualize
it this way, where the first

00:05:39.574 --> 00:05:42.018
term here represents not
this rectangle, but it

00:05:42.018 --> 00:05:43.953
represents this rectangle.

00:05:43.953 --> 00:05:46.017
Notice it has the same
height, but it's just shifted

00:05:46.017 --> 00:05:47.265
over one to the right.

00:05:47.265 --> 00:05:49.072
The second term represents this rectangle.

00:05:49.072 --> 00:05:52.372
This third term represents this rectangle.

00:05:52.392 --> 00:05:53.281
Why does that make sense?

00:05:53.281 --> 00:05:55.760
Well, the area of this first
rectangle, this is going to

00:05:55.760 --> 00:06:00.138
be its height, which is f of
k plus one times its width,

00:06:00.138 --> 00:06:02.767
which is just one, so it's
going to be f of k plus one.

00:06:02.767 --> 00:06:06.118
So the area here is the first
term, the area here is the

00:06:06.118 --> 00:06:08.415
second term, area here is
the third term, the area here

00:06:08.415 --> 00:06:09.727
is the fourth term.

00:06:09.727 --> 00:06:11.671
One way to think about it,
we just shifted all of those

00:06:11.671 --> 00:06:15.624
yellow rectangles one to the right.

00:06:16.162 --> 00:06:18.455
But now this is approximating
a different region.

00:06:18.455 --> 00:06:21.023
This is approximating the
area under the curve not from

00:06:21.023 --> 00:06:24.264
k to infinity, but from
k plus one to infinity.

00:06:24.264 --> 00:06:26.254
And instead of being an
underestimate, it's an

00:06:26.254 --> 00:06:27.598
overestimate.

00:06:27.598 --> 00:06:31.170
Now the curve is contained
within the rectangles.

00:06:31.170 --> 00:06:35.707
So we could say that R sub
k in this context, when we

00:06:35.707 --> 00:06:38.101
conceptualize it this way,
is going to be greater than

00:06:38.101 --> 00:06:40.364
or equal to the improper
integral not from k,

00:06:40.364 --> 00:06:45.364
but from k plus one to infinity f of x dx.

00:06:48.515 --> 00:06:50.332
And what does that allow us to do?

00:06:50.332 --> 00:06:53.091
Well, this places a lower
bound on this, which will

00:06:53.091 --> 00:06:55.003
also place a lower bound on this.

00:06:55.003 --> 00:06:59.779
If this is greater than that,
then this is going to be

00:06:59.779 --> 00:07:02.539
greater than this replaced
with the improper integral.

00:07:02.539 --> 00:07:03.963
So we can write that.

00:07:03.963 --> 00:07:08.226
So S is going to be greater
than or equal to S sub k

00:07:08.226 --> 00:07:13.226
plus the improper integral
from k plus one to infinity

00:07:14.853 --> 00:07:18.872
f of x dx.

00:07:18.981 --> 00:07:20.877
Now, you might be saying,
"Hey, Sal, this looks all

00:07:20.877 --> 00:07:23.898
"crazy, you have all this
abstract notation here, you've

00:07:23.898 --> 00:07:27.903
"introduced an integral sign,
this seems really daunting."

00:07:28.019 --> 00:07:30.480
But as we'll see in the next
few videos, these are actually

00:07:30.480 --> 00:07:33.131
sometimes fairly
straightforward to compute.

00:07:33.131 --> 00:07:34.675
This can be very straightforward
to compute if our

00:07:34.675 --> 00:07:35.843
k is not too large.

00:07:35.843 --> 00:07:37.530
If it's even large, a computer can do it.

00:07:37.530 --> 00:07:40.958
And then these we can actually
often compute sometimes

00:07:40.958 --> 00:07:42.970
numerically, but even more
frequently, and that kind of

00:07:42.970 --> 00:07:45.034
defeats the purpose, but
we can also compute them

00:07:45.034 --> 00:07:49.252
using our analytic tools,
using, I guess you could say,

00:07:49.252 --> 00:07:51.721
the power of calculus.

00:07:51.765 --> 00:07:55.268
And so what this allows us
to do is put a pretty neat

00:07:55.268 --> 00:08:00.268
band around what our actual
value that we converge to is.

00:08:00.584 --> 00:08:03.495
And as we'll see, the
higher our k, the better an

00:08:03.495 --> 00:08:06.832
estimate we get and the
tighter arrange of our

00:08:06.832 --> 00:08:09.047
confidence for that estimate.

00:08:09.047 --> 00:08:11.071
And another way to write
these two inequalities is

00:08:11.071 --> 00:08:13.470
to write a compound inequality
that S is going to be

00:08:13.470 --> 00:08:16.218
less than or equal to this business.

00:08:19.260 --> 00:08:21.499
So copy and paste.

00:08:22.264 --> 00:08:27.264
And S is going to be greater
than or equal to this business.

00:08:29.136 --> 00:08:30.649
So we could say this
business is going to be less

00:08:30.649 --> 00:08:32.340
than or equal to S.

00:08:32.369 --> 00:08:34.436
So let me just copy and paste that.

00:08:34.464 --> 00:08:37.546
So copy and paste.

00:08:37.546 --> 00:08:39.361
Oops, that's not what I wanted to do.

00:08:39.361 --> 00:08:42.556
So copy and paste.

00:08:42.840 --> 00:08:45.173
Then we could write it just like that.

00:08:45.232 --> 00:08:47.720
And so the next series of
videos will actually apply this

00:08:47.720 --> 00:08:50.615
and we'll see that it's
pretty straightforward.

00:08:50.615 --> 00:08:52.644
It looks a little daunting right now.


Radius of convergence using Ratio Test

WEBVTT
Kind: captions
Language: en

00:00:00.148 --> 00:00:02.449
- [Voiceover] Let's say
that we had a function

00:00:02.449 --> 00:00:05.743
defined by the infinite series

00:00:05.743 --> 00:00:09.953
so we're going to from
N equals one to infinity

00:00:09.953 --> 00:00:14.062
and each term is going
to be X to the N plus one

00:00:14.062 --> 00:00:19.062
over four to the N times N to the fourth

00:00:19.348 --> 00:00:20.729
and we could even expand this out.

00:00:20.729 --> 00:00:23.445
When N is equal to one this is going to be

00:00:23.445 --> 00:00:28.445
X squared over four
times one to the fourth

00:00:29.208 --> 00:00:31.384
so which is going to be four.

00:00:31.384 --> 00:00:33.943
When X is equal to two this is going to be

00:00:33.943 --> 00:00:38.173
X to the third power over four squared,

00:00:38.173 --> 00:00:42.637
four squared times two to the fourth power

00:00:42.637 --> 00:00:46.740
and we know this of course is 16 times 16

00:00:46.740 --> 00:00:49.887
which would be 256 but I'll
just leave it like that for now.

00:00:49.887 --> 00:00:53.957
Plus when X is three, this is
going to be X to the fourth

00:00:53.957 --> 00:00:58.957
over four to the fourth power
times four to the fourth

00:00:59.878 --> 00:01:03.062
times four to the fourth
which is really going to be

00:01:03.062 --> 00:01:07.334
four to the eighth power
and we'll just keep going on

00:01:07.334 --> 00:01:09.329
and on and on just to get a sense

00:01:09.329 --> 00:01:11.446
of what this function would look like.

00:01:11.446 --> 00:01:15.109
This function that's
defined by infinite series.

00:01:15.109 --> 00:01:19.428
Now what I'm curious
about is for what X values

00:01:19.428 --> 00:01:24.385
will this function
converge to finite values

00:01:24.385 --> 00:01:26.485
or another way of thinking
about it let's come up

00:01:26.485 --> 00:01:28.444
with the radius of convergence

00:01:28.444 --> 00:01:32.132
or the interval of
convergence for this function.

00:01:32.947 --> 00:01:36.227
Now since we've just been
talking about the ratio test

00:01:36.227 --> 00:01:37.987
you might say, "Well, maybe
the ratio test is useful."

00:01:37.987 --> 00:01:41.325
For what X values is this
thing going to converge

00:01:41.325 --> 00:01:42.916
so let's apply the ratio test.

00:01:42.916 --> 00:01:47.916
Let's take the limit, the
limit as N approaches infinity

00:01:49.026 --> 00:01:54.026
of the absolute value
of the N plus oneth term

00:01:55.747 --> 00:01:58.475
over the Nth term.

00:01:58.475 --> 00:02:01.443
For this particular case
this is going to be the limit

00:02:02.443 --> 00:02:06.911
as N approaches infinity
so the N plus oneth term,

00:02:06.911 --> 00:02:09.361
I'll do this in orange so it'll be

00:02:09.361 --> 00:02:11.983
X to the N plus one plus one.

00:02:11.983 --> 00:02:16.983
So it's going to be X to the N plus two,

00:02:17.297 --> 00:02:21.649
over four to the N plus one,

00:02:21.649 --> 00:02:24.882
times N plus one to the fourth power

00:02:24.882 --> 00:02:28.399
and we're going to divide
that by the Nth term

00:02:28.399 --> 00:02:32.861
which is just going to
be X to the N plus one

00:02:32.861 --> 00:02:37.861
over four to the N times N to the fourth

00:02:38.492 --> 00:02:40.635
which of course is going
to be the same thing

00:02:40.635 --> 00:02:45.340
as the limit as N approaches infinity of,

00:02:45.340 --> 00:02:47.803
I can just copy and paste this, of this

00:02:48.879 --> 00:02:53.879
so copy and paste that times
the reciprocal of this.

00:02:55.333 --> 00:02:58.090
We're dividing by a rational expression,

00:02:58.090 --> 00:03:00.990
that's the same thing as
multiplying by the reciprocal.

00:03:01.728 --> 00:03:06.728
Times four to the N, N to the fourth

00:03:07.186 --> 00:03:11.558
over X to the N plus one power.

00:03:11.558 --> 00:03:14.446
Actually let me just clean
this up a little bit.

00:03:14.446 --> 00:03:16.318
I can re-simplify this.

00:03:16.318 --> 00:03:20.285
Well X to the, let me
get another color here

00:03:20.285 --> 00:03:25.285
so X to the N plus two divided
by X to the N plus one,

00:03:25.919 --> 00:03:29.850
well N plus two minus N plus
one is just going to be one

00:03:29.850 --> 00:03:33.109
so this is just going to simplify
to an X in the numerator.

00:03:33.109 --> 00:03:36.734
Then let's see, four to the N divided by

00:03:36.734 --> 00:03:40.275
four to the N plus one so
this is the same thing as,

00:03:40.275 --> 00:03:42.456
this is the same if we divide
the numerator and denominator

00:03:42.456 --> 00:03:44.839
by four to the N, this is going to be one

00:03:44.839 --> 00:03:47.916
and this is just going to be a four.

00:03:47.916 --> 00:03:50.660
This is going to simplify
to, I don't know,

00:03:50.660 --> 00:03:52.981
I forgot my absolute value
signs, don't want to do that

00:03:52.981 --> 00:03:56.316
so let's keep my absolute value signs here

00:03:56.316 --> 00:03:57.334
the whole time.

00:03:57.334 --> 00:04:00.198
This is going to be equal to the limit

00:04:00.198 --> 00:04:05.198
as N approaches infinity
of the absolute value of,

00:04:06.215 --> 00:04:09.141
we're going to have X, N to
the fourth in the numerator

00:04:09.141 --> 00:04:10.804
so let me write that down.

00:04:10.804 --> 00:04:15.088
We're going to have X times N
to the fourth in the numerator

00:04:15.088 --> 00:04:17.562
and in the denominator
I'm going to have four

00:04:17.562 --> 00:04:19.651
times N plus one to the fourth.

00:04:19.651 --> 00:04:20.820
Well what's that going to be?

00:04:20.820 --> 00:04:25.820
Well if you expand out N
plus one to the fourth,

00:04:25.990 --> 00:04:27.681
it's going to be N to the fourth

00:04:27.681 --> 00:04:29.203
plus a bunch, a bunch of terms

00:04:29.203 --> 00:04:31.614
and a bunch of lower degree terms,

00:04:31.614 --> 00:04:33.038
this is going to be highest degree term

00:04:33.038 --> 00:04:34.145
so if you distribute the four,

00:04:34.145 --> 00:04:35.452
it's going to be four N to the fourth

00:04:35.452 --> 00:04:37.523
plus a bunch of lower degree terms.

00:04:37.523 --> 00:04:40.991
Let me write it that
way so it's over four,

00:04:40.991 --> 00:04:42.767
I'll use the same colors.

00:04:42.767 --> 00:04:46.944
Four N to the fourth,

00:04:46.944 --> 00:04:49.880
plus a bunch of lower degree terms.

00:04:49.880 --> 00:04:53.300
Now, what's the limit as N
approaches infinity here?

00:04:53.300 --> 00:04:54.667
Well, we have the same,

00:04:54.667 --> 00:04:56.674
we want to look at the
highest degree terms.

00:04:56.674 --> 00:04:58.011
We want to look at the N to the fourth,

00:04:58.011 --> 00:04:59.475
the fourth degree terms

00:04:59.475 --> 00:05:01.778
and so it's really going to
be the coefficient on this

00:05:01.778 --> 00:05:03.774
and one way to think about it
if you divide the numerator

00:05:03.774 --> 00:05:06.115
and the denominator
both by N to the fourth,

00:05:06.115 --> 00:05:07.760
these are the only terms,

00:05:07.760 --> 00:05:09.838
you're going to have an X in the numerator

00:05:09.838 --> 00:05:11.181
and the denominator
you're going to have four

00:05:11.181 --> 00:05:13.061
and all these others
terms are going to divided

00:05:13.061 --> 00:05:14.501
by some power of N.

00:05:14.501 --> 00:05:16.749
As your limit approaches,
as N approaches infinity

00:05:16.749 --> 00:05:17.953
all the other terms are going to zero

00:05:17.953 --> 00:05:18.822
and all you're going to have left

00:05:18.822 --> 00:05:21.582
in the denominator is a
four and X in the numerator.

00:05:21.582 --> 00:05:25.676
This right over here, the
limit as N approaches infinity

00:05:25.676 --> 00:05:27.402
is going to be X over four

00:05:27.402 --> 00:05:30.226
or I should say the absolute
value of X over four.

00:05:31.026 --> 00:05:33.682
It's going to be equal
to the absolute value

00:05:33.682 --> 00:05:38.682
of X over the absolute
value of X over four.

00:05:39.587 --> 00:05:42.726
Now we can apply the ratio test

00:05:42.726 --> 00:05:44.065
or I guess we're always applying him

00:05:44.065 --> 00:05:46.676
but now we can really test,
do the test part of it

00:05:46.676 --> 00:05:48.379
and the ratio test tells us

00:05:48.379 --> 00:05:50.178
that if this limit right over here,

00:05:50.178 --> 00:05:54.435
as long as it evaluates to something

00:05:54.435 --> 00:05:59.435
that is less than one, then
our series will converge.

00:06:00.936 --> 00:06:03.626
We just have to say that this
expression right over here

00:06:03.626 --> 00:06:04.884
has to be less than one.

00:06:05.899 --> 00:06:08.789
Our function is going to
converge to finite values

00:06:08.789 --> 00:06:10.785
as long as the absolute
value of X over four

00:06:10.785 --> 00:06:15.785
is less than one and this is
dividing the absolute value of,

00:06:16.481 --> 00:06:17.910
or the absolute value of X over four

00:06:17.910 --> 00:06:20.083
is going to be the same thing
as the absolute value of X

00:06:20.083 --> 00:06:22.997
divided by four so it's four is positive

00:06:22.997 --> 00:06:24.576
so that just has to be less than one.

00:06:24.576 --> 00:06:26.575
Multiply both sides by four,

00:06:26.575 --> 00:06:28.694
won't change the equality
since four is positive

00:06:28.694 --> 00:06:30.515
so you're going to have
the absolute value of X

00:06:30.515 --> 00:06:32.545
has to be less than four.

00:06:32.545 --> 00:06:36.522
Just like that we have defined
the radius of convergence.

00:06:36.522 --> 00:06:39.695
This right over here is
the radius of convergence

00:06:39.695 --> 00:06:42.107
for our function or for
our infinite series,

00:06:42.107 --> 00:06:45.143
radius of convergence.

00:06:45.143 --> 00:06:47.327
Or if we wanted to write
that same information

00:06:47.327 --> 00:06:50.110
as an interval of
conversions, we would say

00:06:50.110 --> 00:06:54.706
that X has to be less than four

00:06:54.706 --> 00:06:56.800
or greater than negative four.

00:06:56.800 --> 00:06:58.877
This is the same
information right over here,

00:06:58.877 --> 00:07:03.877
express this way, this is
the interval of convergence.

00:07:09.009 --> 00:07:12.852
These are the X values for which
this original function here

00:07:12.852 --> 00:07:16.667
is going to converge to finite values.


Limit comparison test

WEBVTT
Kind: captions
Language: en

00:00:00.502 --> 00:00:02.970
- Let's remind ourselves,
give ourselves a review

00:00:02.970 --> 00:00:05.853
of the Comparison Test,
see where it can be useful,

00:00:05.853 --> 00:00:08.972
and maybe see where it might
not be so useful, but luckily

00:00:08.972 --> 00:00:11.569
we'll also see the Limit Comparison Test,

00:00:11.569 --> 00:00:15.520
which can be applicable in a
broader category of situations.

00:00:15.520 --> 00:00:18.053
So we've already seen
this, we want to prove

00:00:18.053 --> 00:00:20.748
that the infinite series from
n equals one to infinity of

00:00:20.748 --> 00:00:24.594
one over two to the n plus one
converges how can we do that?

00:00:24.594 --> 00:00:28.025
Well, each of these terms are
greater than or equal to zero,

00:00:28.056 --> 00:00:30.179
and we can construct another series where

00:00:30.179 --> 00:00:32.255
each of the corresponding
terms are greater than

00:00:32.255 --> 00:00:34.151
each of these corresponding terms.

00:00:34.151 --> 00:00:37.603
And that other series,
the one that jumps out at,

00:00:37.603 --> 00:00:40.901
that will likely jump out
for most folks would be

00:00:40.901 --> 00:00:45.118
one over two to the n,
one over two to the n is

00:00:45.118 --> 00:00:50.118
greater than, is greater
than, and all we would

00:00:50.651 --> 00:00:53.484
have to really say is
greater than or equal to, but

00:00:53.484 --> 00:00:55.093
we could actually explicitly say it's, oh

00:00:55.093 --> 00:00:57.110
I'll just write it's
greater than or equal to,

00:00:57.110 --> 00:01:01.119
one over two to the n plus one for,

00:01:01.119 --> 00:01:06.085
n is equal to one, two, all
the way to infinity - why?

00:01:06.085 --> 00:01:08.085
Because this denominator
is always going to be

00:01:08.085 --> 00:01:10.719
greater by one if you're
denominator is greater,

00:01:10.719 --> 00:01:15.602
the overall expression
is going to be less,

00:01:15.602 --> 00:01:18.719
and because of that, because
each of these terms are,

00:01:18.719 --> 00:01:22.836
they're all positive, this
one, each corresponding term is

00:01:22.836 --> 00:01:26.267
greater than that one, and
by the Comparison Test,

00:01:26.267 --> 00:01:28.700
because this one converges,
this kinda provides an

00:01:28.700 --> 00:01:31.602
upper bound, because this series
we already know converges,

00:01:31.602 --> 00:01:35.368
we can say, so because this one converges,

00:01:35.368 --> 00:01:38.652
we can say that this one converges.

00:01:38.652 --> 00:01:41.650
Now, let's see if we can
apply a similar logic

00:01:41.650 --> 00:01:44.532
to a slightly different series.

00:01:44.532 --> 00:01:46.233
Let's say we have the series

00:01:46.233 --> 00:01:49.826
the sum from n equals one to infinity of

00:01:49.826 --> 00:01:52.730
one over two to the n minus one.

00:01:52.730 --> 00:01:54.863
In this situation can we do just

00:01:54.863 --> 00:01:58.030
the straight up Comparison Test?

00:01:58.030 --> 00:02:00.562
Well no, because you cannot say that

00:02:00.562 --> 00:02:03.447
one over two to the n is
greater than or equal to

00:02:03.447 --> 00:02:05.573
one over two to the n minus one.

00:02:05.573 --> 00:02:07.662
Here, the denominator is lower, means

00:02:07.662 --> 00:02:10.107
the expression is greater,
means that this can't,

00:02:10.107 --> 00:02:12.940
each of these terms can't provide
an upper bound on this one.

00:02:12.940 --> 00:02:15.138
This one is a little bit larger,

00:02:15.138 --> 00:02:17.307
but on the other hand you're
like, 'Ok, I get that.'

00:02:17.307 --> 00:02:19.939
But look, as n gets large,
the two to the n is going to

00:02:19.939 --> 00:02:23.724
really dominate the minus
one or the plus one, or

00:02:23.724 --> 00:02:26.724
this one has nothing there,
this one just has two to the n.

00:02:26.724 --> 00:02:28.591
The two to the n is
really going to describe

00:02:28.591 --> 00:02:30.591
the behavior of what this thing does.

00:02:30.591 --> 00:02:32.756
And I would agree with you, but we just

00:02:32.756 --> 00:02:35.474
haven't proven that it actually works,

00:02:35.474 --> 00:02:39.092
and that's where the Limit
Comparison Test comes in helpful.

00:02:39.092 --> 00:02:44.092
So let me write that down,
'Limit, Limit Comparison Test,

00:02:45.641 --> 00:02:50.093
'Limit Comparison Test',
and I'll write it down

00:02:50.093 --> 00:02:52.109
a little bit formally,
but then we'll apply it to

00:02:52.109 --> 00:02:55.120
this infinite series right over here.

00:02:55.120 --> 00:02:58.318
So what Limit Comparison Test tells us,

00:02:58.318 --> 00:03:01.543
that if I have two infinite series,

00:03:01.543 --> 00:03:06.474
so this is going from n equals
k to infinity, of a sub n,

00:03:06.474 --> 00:03:07.525
I'm not going to prove it here,

00:03:07.525 --> 00:03:09.792
we'll just learn to apply it first.

00:03:09.792 --> 00:03:14.394
This one goes from n equals
k to infinity, of b sub n.

00:03:14.394 --> 00:03:18.761
And we know that each of
the terms, a sub n, are

00:03:18.761 --> 00:03:20.728
greater than or equal to zero,

00:03:20.728 --> 00:03:23.428
and we know each of
the terms, b sub n, are

00:03:23.428 --> 00:03:25.935
actually we're just going
to say, greater than zero,

00:03:25.935 --> 00:03:26.997
because it's actually
going to show up in the

00:03:26.997 --> 00:03:28.281
denominator of an expression,

00:03:28.281 --> 00:03:30.213
so we don't want it to be equal to zero,

00:03:30.213 --> 00:03:32.313
for all the n's that we care about.

00:03:32.313 --> 00:03:37.313
So for all n equal to k, k plus one,

00:03:38.359 --> 00:03:41.804
k plus two, on and on, and on and on, and,

00:03:41.804 --> 00:03:44.314
and this is the key, this
is where the limit of

00:03:44.314 --> 00:03:47.466
the Limit Comparison Test comes into play,

00:03:47.466 --> 00:03:52.466
and, if the limit, the limit
as n approaches infinity,

00:03:55.674 --> 00:04:00.674
of a sub n over b sub n, b sub n

00:04:01.972 --> 00:04:05.281
is positive and finite,
is positive and finite,

00:04:09.885 --> 00:04:12.890
that either both series converge,

00:04:12.890 --> 00:04:16.698
or both series diverge,
so let me write that.

00:04:16.698 --> 00:04:20.536
So then, that tells us that either,

00:04:20.536 --> 00:04:24.520
either both series converge,

00:04:25.458 --> 00:04:30.458
or both diverge, which
is really really useful.

00:04:30.719 --> 00:04:32.768
It's kind of a more
formal way of saying that,

00:04:32.768 --> 00:04:36.220
'Hey look, as n approaches
infinity, if these have similar

00:04:36.220 --> 00:04:37.779
'behaviors then they are either going to

00:04:37.779 --> 00:04:40.102
'converge, or they're
both going to diverge.'

00:04:40.102 --> 00:04:42.473
Let's apply that right over here.

00:04:42.473 --> 00:04:45.777
Well if we say that our b sub n

00:04:45.777 --> 00:04:47.836
is one over two to the n, just like

00:04:47.836 --> 00:04:49.664
we did up there, one over two to the n,

00:04:49.664 --> 00:04:51.864
so we're going to compare, so these

00:04:51.864 --> 00:04:54.422
two series right over
here, notice it satisfies

00:04:54.422 --> 00:04:58.301
all of these constraints,
so let's take the limit,

00:04:58.301 --> 00:05:01.864
the limit as n approaches infinity

00:05:01.864 --> 00:05:05.187
of a sub over b sub n, so it's going to be

00:05:05.187 --> 00:05:10.187
one over two to the n minus one over

00:05:10.288 --> 00:05:15.288
one over two to the n, and
what's that going to be equal to?

00:05:16.723 --> 00:05:18.458
Well that's going to be equal to

00:05:18.458 --> 00:05:23.098
the limit as n approaches infinity, of

00:05:23.098 --> 00:05:25.617
two, if you divide by
one over two to the n

00:05:25.617 --> 00:05:27.426
that's just going to be the
same thing as multiplying

00:05:27.426 --> 00:05:28.962
by two to the n, so it's going to be

00:05:28.962 --> 00:05:33.893
two to the n over two to the n minus one.

00:05:34.960 --> 00:05:37.856
Over two to the n minus
one, and this clearly,

00:05:37.856 --> 00:05:39.605
what's happening in the
numerator and the denominator,

00:05:39.605 --> 00:05:43.055
these are approaching the
same, well we can even

00:05:43.055 --> 00:05:45.096
write it like, we can
even write it like this,

00:05:45.096 --> 00:05:47.002
divide the numerator and
denomiator by two to the n,

00:05:47.002 --> 00:05:48.602
if you want, although it's probably

00:05:48.602 --> 00:05:50.274
going to jump out at you at this point.

00:05:51.181 --> 00:05:56.181
So, limit as n approaches infinity,

00:05:56.447 --> 00:05:58.063
let me just scroll over
to the right a little bit.

00:05:58.063 --> 00:06:00.199
If I divide the numerator by two to the n,

00:06:00.199 --> 00:06:01.996
I'm just going to have one.

00:06:01.996 --> 00:06:04.064
If I divide the denominator
by two to the n,

00:06:04.064 --> 00:06:09.064
I'm going to have one minus
one over two to the n.

00:06:09.325 --> 00:06:11.624
And now it becomes clear,
this thing right over here

00:06:11.624 --> 00:06:13.245
is just going to go to zero, and you're

00:06:13.245 --> 00:06:15.112
going to have one over one.

00:06:15.112 --> 00:06:16.945
The important thing is that

00:06:16.945 --> 00:06:21.530
this limit is positive and
finite because this thing is

00:06:21.530 --> 00:06:26.530
so this thing right over
here, is positive and finite,

00:06:26.911 --> 00:06:30.941
the limit is one is positive and finite,

00:06:30.941 --> 00:06:33.675
if this thing converges and
this thing converges, or

00:06:33.675 --> 00:06:35.826
this thing diverges, and this diverges,

00:06:35.826 --> 00:06:39.357
well, we already know
this thing converges,

00:06:39.357 --> 00:06:41.391
it's a geometric series where the

00:06:41.391 --> 00:06:45.091
common ratio is less than
one, and so therefore

00:06:45.091 --> 00:06:50.091
this must converge as well,
so that converges as well.