Leveraging properties of series to find sum

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.690
Let's try to find the sum
of this right over here,

00:00:04.690 --> 00:00:07.360
or let's try to evaluate this
expression right over here.

00:00:07.360 --> 00:00:11.874
So we're evaluating what
this sum turns out to be.

00:00:11.874 --> 00:00:13.540
Now, there's a bunch
of ways to do this.

00:00:13.540 --> 00:00:15.150
You could literally just
do it by brute force.

00:00:15.150 --> 00:00:16.858
You could say, well,
what does this equal

00:00:16.858 --> 00:00:20.580
when n equals 1, when n equals
2, all the way to n equals 7?

00:00:20.580 --> 00:00:22.670
And that would be
completely legitimate.

00:00:22.670 --> 00:00:24.250
But I view this
is an opportunity

00:00:24.250 --> 00:00:28.130
to look at some properties
of sigma notation.

00:00:28.130 --> 00:00:29.319
So let's split this out.

00:00:29.319 --> 00:00:30.860
So the first thing
that you might say

00:00:30.860 --> 00:00:32.790
is, well, look, if
I were to sum up

00:00:32.790 --> 00:00:37.570
all of these when n equals 1
and n equals 2, all the way to n

00:00:37.570 --> 00:00:39.740
equals 7, it's reasonable
that this is going

00:00:39.740 --> 00:00:46.660
to be the same thing as the
sum of 3 n squared from n

00:00:46.660 --> 00:00:55.460
equals 1 to 7 plus the
sum from n equals 1 to 7

00:00:55.460 --> 00:00:57.940
of 2n squared-- of
2n, I should say,

00:00:57.940 --> 00:01:06.690
that right over there, plus the
sum from n equals 1 to 7 of 4.

00:01:06.690 --> 00:01:09.490
And if you find this a
little bit confusing,

00:01:09.490 --> 00:01:11.970
I encourage you to expand
to both of these things

00:01:11.970 --> 00:01:14.640
out and realize that when
you rearrange the terms,

00:01:14.640 --> 00:01:16.340
that you will get
these two things.

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I'm not doing a
rigorous proof here,

00:01:18.080 --> 00:01:19.996
but hopefully, if you
were to expand this out,

00:01:19.996 --> 00:01:22.210
you'd see that this is
not an unreasonable thing

00:01:22.210 --> 00:01:24.390
to claim right over here.

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Now, out of all of
these, this last piece

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is pretty easy to evaluate.

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When n equals 1, this
thing is equal to 4.

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When n equals 2, this
thing is equal to 4.

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When n equals 3, this
thing is equal to 4.

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So you're essentially going
to take seven 4's and add them

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together.

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So this is essentially just
going to evaluate to 7 times

00:01:43.460 --> 00:01:46.730
4, or 28.

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Now let's look at this
piece right over here.

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Now, once again, we can
just do it by brute force.

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2 times 1 is 2 plus
2 times 2 is 4.

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So you're going to
essentially do the first seven

00:01:57.380 --> 00:02:00.660
multiples of 2 is one way
you could think about this.

00:02:00.660 --> 00:02:03.075
Or, if we were to expand
it out-- actually,

00:02:03.075 --> 00:02:03.950
let me expand it out.

00:02:03.950 --> 00:02:12.870
This is going to be 2 plus
4 plus 6, all the way to,

00:02:12.870 --> 00:02:15.530
when this is 7,
all the way to 14.

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You could factor out a 2.

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And so this is going to
become 2 times 1 plus 2

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plus 3 all the way to 7.

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And so you can rewrite
this piece right over

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here as 2 times the sum--
so we're essentially just

00:02:30.230 --> 00:02:33.660
factoring out the 2--
2 times the sum, which

00:02:33.660 --> 00:02:37.650
is the sum from n
equals 1 to 7 of n.

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So this is this piece.

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We still have this 28
that we have to add.

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So we have this 28.

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And we draw the
parentheses so you

00:02:44.640 --> 00:02:47.760
don't think that the 28 is
part of this right over here.

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And now we can do the
same thing with this.

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3 times n-- we're taking from
n equals 1 to 7 of 3 n squared.

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Doing the same exact thing
as we just did in magenta,

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this is going to be equal
to 3 times the sum from n

00:03:03.710 --> 00:03:07.390
equals 1 to 7 of n squared.

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We're essentially
factoring out the 3.

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We're factoring out the 2.

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n squared.

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And once again, we can
put parentheses just

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to clarify things.

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Now, at this point, there
are formulas to evaluate

00:03:21.390 --> 00:03:22.570
each of these things.

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There's a formula to evaluate
this thing right over here.

00:03:25.410 --> 00:03:27.679
There's a formula to evaluate
this thing over here.

00:03:27.679 --> 00:03:28.720
And you can look them up.

00:03:28.720 --> 00:03:30.640
And actually, I'll
give you the formulas,

00:03:30.640 --> 00:03:32.350
in case you're curious.

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This formula, one
expression of this formula

00:03:35.340 --> 00:03:39.790
is that this is going to
be n to the third over 3

00:03:39.790 --> 00:03:45.000
plus n squared over
2 plus n over 6.

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That's one formula for that.

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And one formula for this piece
right over here, going from n

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equals 1 to 7-- sorry.

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Let me make it clear.

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This n is actually what your
terminal value should be.

00:03:59.040 --> 00:04:02.850
So this should be 7 to
the third power over 3--

00:04:02.850 --> 00:04:04.250
so it's not this n.

00:04:04.250 --> 00:04:08.720
I was just mindlessly using the
formula-- 7 to the third over 3

00:04:08.720 --> 00:04:13.890
plus 7 squared over 2 plus 7/6.

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So that's this sum.

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And this sum, you could
view it as the average

00:04:17.790 --> 00:04:19.240
of the first and the last terms.

00:04:19.240 --> 00:04:20.610
So the first term is 1.

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The last term is 7.

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So take their average
and then multiply it

00:04:24.770 --> 00:04:26.990
times the number
of terms you have.

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So times-- you have 7 terms.

00:04:31.360 --> 00:04:34.240
So what is this middle
one going to evaluate to?

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Well, 1 times-- and of course,
we have this 2 out front.

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This green is just this
part right over here.

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So you have 2 times this.

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And over here, you have 3 times
this business right over here.

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So if we evaluate this
one, 2 times-- let's see.

00:04:51.180 --> 00:04:53.700
1 plus 7 is 8,
divided by 2 is 4.

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4 times 2 is 8.

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Times 7, it's 56.

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So that becomes 56.

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Now, this-- let's see.

00:05:01.390 --> 00:05:04.600
This is actually-- well, we
could evaluate this if we want.

00:05:04.600 --> 00:05:06.480
And I guess we could
take out a calculator

00:05:06.480 --> 00:05:08.479
if we wanted to figure
out 7 to the third power.

00:05:08.479 --> 00:05:11.620
Actually, let's just do
that just to save time here.

00:05:11.620 --> 00:05:13.460
So let's calculate.

00:05:13.460 --> 00:05:18.010
So we have 7 to the
third power divided by 3

00:05:18.010 --> 00:05:24.910
plus 7 squared divided
by 2 plus 7 divided by 6

00:05:24.910 --> 00:05:28.470
gives us a drum roll of 140.

00:05:28.470 --> 00:05:33.180
So this is going to
be equal to 3 times

00:05:33.180 --> 00:05:42.330
140-- let me do it in that
color-- 3 times 140 plus 56

00:05:42.330 --> 00:05:44.840
plus 28.

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And since we get our calculator
out, let's just use it.

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So let's see.

00:05:48.430 --> 00:05:57.770
140 times 3 is 420,
of course, plus 56,

00:05:57.770 --> 00:06:05.580
plus 28-- we deserve a drum
roll now-- gets us to 504.

00:06:05.580 --> 00:06:11.112
So this sum right over
here is equal to 504.

00:06:11.112 --> 00:06:13.330
And once again, multiple
ways you can do it.

00:06:13.330 --> 00:06:16.450
But it's nice to know
that there are these ways

00:06:16.450 --> 00:06:18.046
to break down the problem.

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And there are all
these formulas.

00:06:19.420 --> 00:06:23.340
Now, I encourage you
to look at the formulas

00:06:23.340 --> 00:06:25.699
to see how this is actually
derived and proved.

00:06:25.699 --> 00:06:27.240
I'm not a big fan
of just saying, oh,

00:06:27.240 --> 00:06:28.364
there's a formula for this.

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You just apply it.

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The formula here is
whatever this terminal value

00:06:32.000 --> 00:06:34.400
is to the third power over
3 plus that squared over 2,

00:06:34.400 --> 00:06:36.760
plus that over 6.

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I encourage you to
look up on our site,

00:06:40.740 --> 00:06:44.350
on Khan Academy, the formula
for the sum of n squares,

00:06:44.350 --> 00:06:46.380
and it'll tell you where
this is derived from.

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And also, the formula for the
sum of an arithmetic series,

00:06:49.540 --> 00:06:53.620
and it'll tell you where
this is derived from.


Alternate formula for sum of n squares

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.720
In the last video,
we saw that if we

00:00:02.720 --> 00:00:12.540
were to take the sum from i
equals 0 to n of i squared,

00:00:12.540 --> 00:00:15.240
that this could be equal to--
and we did it in two videos.

00:00:15.240 --> 00:00:17.380
It was a little bit hairy,
but we powered through.

00:00:17.380 --> 00:00:21.680
This is equal to 1/3
n to the third power.

00:00:21.680 --> 00:00:23.970
Let me write it a little
cleaner than that.

00:00:23.970 --> 00:00:32.140
n to the third power plus
1/2 n squared plus 1/6 n.

00:00:32.140 --> 00:00:35.810
And that is absolutely
true, but you

00:00:35.810 --> 00:00:37.280
might have seen a
different formula

00:00:37.280 --> 00:00:38.670
for this that doesn't
look like that.

00:00:38.670 --> 00:00:40.750
And so what I want to do is now
manipulate this a little bit

00:00:40.750 --> 00:00:43.080
algebraically to come up
with another formula that's

00:00:43.080 --> 00:00:45.750
typically seen for this sum.

00:00:45.750 --> 00:00:47.650
So the first thing
that we might realize

00:00:47.650 --> 00:00:53.390
is, well, what happens
if we factor out a 1/6 n?

00:00:53.390 --> 00:00:55.840
So this is going to
be equal to-- let's

00:00:55.840 --> 00:00:58.750
factor out this right over here.

00:00:58.750 --> 00:01:04.989
So that would be equal to
1/6 n times-- so let's see.

00:01:04.989 --> 00:01:09.630
1/3 divided by 1/6, that's
the same thing as 1/3 times 6.

00:01:09.630 --> 00:01:14.864
So this right over here is going
to be equal to 2 n squared.

00:01:14.864 --> 00:01:15.780
And you can verify it.

00:01:15.780 --> 00:01:18.750
1/6 times 2 is 2/6 or 1/3.

00:01:18.750 --> 00:01:22.200
n times n squared is
n to the third power.

00:01:22.200 --> 00:01:24.820
Now, 1/2 divided by
1/6 is the same thing

00:01:24.820 --> 00:01:27.790
as 1/2 times 6, which is 3.

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So this term right over here
is going to be plus 3 times n.

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And then finally, you
have this term right

00:01:38.130 --> 00:01:41.430
over here, which is just
going to be equal to 1.

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And now we're faced with
factoring this thing

00:01:45.930 --> 00:01:47.470
right over here.

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Now, you could to do
this-- well, let's factor

00:01:52.080 --> 00:01:53.640
by grouping this.

00:01:53.640 --> 00:01:55.880
And remember, when you
factor by grouping,

00:01:55.880 --> 00:01:58.910
you want to
essentially break this

00:01:58.910 --> 00:02:05.580
into two numbers where the
product of those two numbers

00:02:05.580 --> 00:02:10.759
is equal to the
product of 2 times 1.

00:02:10.759 --> 00:02:13.840
And the obvious thing
is, well, 2 times 1 is 2.

00:02:13.840 --> 00:02:18.460
If I break this into 2 and 1,
if I break this 3 into 2 and 1,

00:02:18.460 --> 00:02:21.690
well, 2 times 1 is
definitely 2 times 1.

00:02:21.690 --> 00:02:22.880
So let me rewrite this.

00:02:22.880 --> 00:02:25.310
I'm just factoring by grouping
here, nothing super fancy.

00:02:25.310 --> 00:02:37.047
So if I rewrite this as 2 n
squared plus n plus 2n plus 1--

00:02:37.047 --> 00:02:38.880
so this is all the stuff
in this parentheses

00:02:38.880 --> 00:02:42.320
right over here-- well,
this part right over here

00:02:42.320 --> 00:02:43.582
we can factor out an n.

00:02:43.582 --> 00:02:46.040
So this is going to be equal
to-- let me just write the 1/6

00:02:46.040 --> 00:02:49.310
n here so you know
what I'm doing.

00:02:49.310 --> 00:02:51.870
So if we factor out
an n right over here,

00:02:51.870 --> 00:03:01.180
we would have n times 2n plus 1.

00:03:05.720 --> 00:03:08.090
So that's what's in
green right over here.

00:03:08.090 --> 00:03:10.240
That's this part
right over here.

00:03:10.240 --> 00:03:14.740
And then what I have
here, I could write this

00:03:14.740 --> 00:03:21.220
as plus 1 times 2n plus 1.

00:03:21.220 --> 00:03:27.230
Let me do it in that same
color, so 1 times 2n plus 1.

00:03:30.910 --> 00:03:35.010
And so now, you can
factor out a 2n plus 1.

00:03:35.010 --> 00:03:40.910
And this thing simplifies
to 1/6 n times 2n plus 1.

00:03:40.910 --> 00:03:44.740
I'm just factoring
out a 2n plus 1.

00:03:44.740 --> 00:03:48.980
And then if you factor a 2n plus
1 out of each of these terms,

00:03:48.980 --> 00:03:51.290
you're going to
just an n and a 1.

00:03:51.290 --> 00:04:03.490
So 1/6 times n times 2n
plus 1 times n plus 1.

00:04:07.699 --> 00:04:09.990
And if this was a little bit
confusing what I did here,

00:04:09.990 --> 00:04:11.240
I encourage you to
review the videos

00:04:11.240 --> 00:04:12.365
on factoring by grouping.

00:04:15.490 --> 00:04:17.880
And another way we
can rewrite this,

00:04:17.880 --> 00:04:20.310
we could write this whole
thing as being equal to--

00:04:20.310 --> 00:04:22.101
and maybe I'll write
it in this green color

00:04:22.101 --> 00:04:23.210
that we started off with.

00:04:23.210 --> 00:04:26.780
This is the same
thing as-- and I'll

00:04:26.780 --> 00:04:33.810
write it as n times
n plus 1 times 2n

00:04:33.810 --> 00:04:38.880
plus 1, all of that over 6.

00:04:38.880 --> 00:04:41.780
So this expression is
equivalent to this expression,

00:04:41.780 --> 00:04:44.690
and it's equivalent
to this value

00:04:44.690 --> 00:04:46.810
right over here, so
whichever one you

00:04:46.810 --> 00:04:50.690
find to be a little bit
easier to think about.


Finding the sum of n squares part 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.870
We're now ready to
solve for A, B, and C,

00:00:04.870 --> 00:00:08.020
given these three equations
with three unknowns.

00:00:08.020 --> 00:00:10.700
So let's first start with
these top two constraints.

00:00:10.700 --> 00:00:12.930
So I have this one
right over here in blue.

00:00:12.930 --> 00:00:16.070
Let me just copy and
paste it right over here.

00:00:16.070 --> 00:00:18.070
And then let me rewrite
this constraint,

00:00:18.070 --> 00:00:20.840
but let me multiply
it by negative 2

00:00:20.840 --> 00:00:23.140
so that I can cancel
out the C terms.

00:00:23.140 --> 00:00:28.530
So it's negative 2A
minus 2B minus 2C

00:00:28.530 --> 00:00:30.510
is equal to negative 2.

00:00:30.510 --> 00:00:37.200
And then, if I add the left-hand
sides and the right-hand sides,

00:00:37.200 --> 00:00:44.540
on the left-hand sides, I get
6A plus 2B-- these cancel out--

00:00:44.540 --> 00:00:47.420
is equal to 3.

00:00:47.420 --> 00:00:49.150
And then the other
constraint I could use

00:00:49.150 --> 00:00:51.840
is this orange one
and the green to get

00:00:51.840 --> 00:00:53.670
the C's to cancel out here.

00:00:53.670 --> 00:01:05.170
So I have 27A plus 9B
plus 3C is equal to 14.

00:01:05.170 --> 00:01:06.210
And here, let's see.

00:01:06.210 --> 00:01:07.710
I can multiply it
by negative 3 if I

00:01:07.710 --> 00:01:09.376
want to get the C's
to cancel out again.

00:01:09.376 --> 00:01:13.240
And my whole goal is to have two
equations and two unknowns that

00:01:13.240 --> 00:01:17.240
have leveraged these two
constraints in tandem

00:01:17.240 --> 00:01:19.000
and these two
constraints in tandem.

00:01:19.000 --> 00:01:20.660
So now I'm going do these two.

00:01:20.660 --> 00:01:22.770
So I've multiplied this
equation times negative 3.

00:01:22.770 --> 00:01:30.490
I get negative 3A minus 3B
minus 3C is equal to negative 3.

00:01:30.490 --> 00:01:33.630
And now I can perform
the subtraction.

00:01:33.630 --> 00:01:40.450
And I get 24A plus 6B-- these
cancel-- is equal to 11.

00:01:40.450 --> 00:01:43.310
Now I have two equations
in two unknowns.

00:01:43.310 --> 00:01:44.010
And let's see.

00:01:44.010 --> 00:01:46.470
If I multiply this
equation right

00:01:46.470 --> 00:01:49.320
over here times
negative 3, I should

00:01:49.320 --> 00:01:50.820
get the B's to cancel out.

00:01:50.820 --> 00:01:52.290
So let's do that.

00:01:52.290 --> 00:01:54.350
So if I multiply this
times negative 3,

00:01:54.350 --> 00:01:59.250
this constraint, or this
term, 6A times negative 3,

00:01:59.250 --> 00:02:02.290
is negative 18A.

00:02:02.290 --> 00:02:07.530
2B times negative 3, minus 6B,
is equal to 3 times negative 3

00:02:07.530 --> 00:02:09.100
is equal to negative 9.

00:02:09.100 --> 00:02:15.417
And now if we add both
sides, on the left-hand side,

00:02:15.417 --> 00:02:16.250
we have-- let's see.

00:02:16.250 --> 00:02:24.610
24A minus 18A is 6A-- these
cancel-- is equal to 11 minus 9

00:02:24.610 --> 00:02:25.750
is 2.

00:02:25.750 --> 00:02:27.910
Divide both sides by 6.

00:02:27.910 --> 00:02:34.690
We get A is equal to 2/6,
which is the same thing as 1/3.

00:02:34.690 --> 00:02:38.430
And so now we can substitute
back to solve for B.

00:02:38.430 --> 00:02:39.190
So let's see.

00:02:39.190 --> 00:02:44.360
We have 6 times 1/3.

00:02:44.360 --> 00:02:46.610
Our A is 1/3.

00:02:46.610 --> 00:02:50.860
Plus 2B is equal to 3.

00:02:50.860 --> 00:02:53.452
6 times 1/3 is 2.

00:02:53.452 --> 00:02:55.940
2 plus 2B is equal to 3.

00:02:55.940 --> 00:02:58.100
Subtract 2 from both sides.

00:02:58.100 --> 00:03:00.860
2B is equal to 1.

00:03:00.860 --> 00:03:02.720
Divide both sides by 2.

00:03:02.720 --> 00:03:05.470
B is equal to 1/2.

00:03:05.470 --> 00:03:06.910
So A is 1/3.

00:03:06.910 --> 00:03:08.000
B is 1/2.

00:03:08.000 --> 00:03:10.190
Now we just have to solve for C.

00:03:10.190 --> 00:03:12.200
So we can go back to
this original equation

00:03:12.200 --> 00:03:13.450
right over here.

00:03:13.450 --> 00:03:20.800
So we have 1/3 plus 1/2
plus C is equal to 1.

00:03:20.800 --> 00:03:24.110
Actually, let me do that over
here so I have some space.

00:03:24.110 --> 00:03:26.940
So I have some space, let
me do it right over here.

00:03:26.940 --> 00:03:32.975
So I have 1/3 plus 1/2
plus C is equal to 1.

00:03:32.975 --> 00:03:35.280
Now we find ourselves
a common denominator.

00:03:35.280 --> 00:03:36.200
So let's see.

00:03:36.200 --> 00:03:38.875
A common multiple
of 3, 2, and-- I

00:03:38.875 --> 00:03:42.050
guess you could say 1-- this
is 1 over 1-- is going to be 6.

00:03:42.050 --> 00:03:46.230
So I can rewrite
this as 2/6 plus 3/6

00:03:46.230 --> 00:03:49.340
plus C is equal to 6/6.

00:03:49.340 --> 00:03:51.440
1 is the same thing as 6/6.

00:03:51.440 --> 00:03:56.430
So this is 5/6 plus
C is equal to 6/6.

00:03:56.430 --> 00:03:59.460
Subtract 5/6 from both sides.

00:03:59.460 --> 00:04:03.040
We get C is equal to 1/6.

00:04:03.040 --> 00:04:06.290
So C is equal to 1/6.

00:04:06.290 --> 00:04:07.016
And there we are.

00:04:07.016 --> 00:04:08.140
We deserve a drum roll now.

00:04:08.140 --> 00:04:16.079
We've figured out a formula for
the sum of the first n squares.

00:04:16.079 --> 00:04:17.470
So we can rewrite this.

00:04:17.470 --> 00:04:20.920
This formula is now
going to be-- A is 1/3.

00:04:20.920 --> 00:04:30.020
So it's 1/3 n to the third
power plus 1/2 n squared--

00:04:30.020 --> 00:04:33.270
let me make sure it doesn't
look like that n is a-- 1/2

00:04:33.270 --> 00:04:39.880
n squared plus 1/6 n.

00:04:39.880 --> 00:04:46.600
So this is a pretty
handy formula.

00:04:46.600 --> 00:04:50.050
If now you wanted to find
0 squared plus 1 squared

00:04:50.050 --> 00:04:52.640
plus 2 squared plus 3 squared,
all the way to 100 squared,

00:04:52.640 --> 00:04:56.260
instead of squaring 100 numbers
and then adding them together,

00:04:56.260 --> 00:04:58.630
you could figure out 1/3
of 100 to the third power

00:04:58.630 --> 00:05:02.331
plus 1/2 times 100 squared
plus 1/6 times 100.

00:05:02.331 --> 00:05:03.830
Or you could do
that for any number.

00:05:03.830 --> 00:05:10.030
So this right over here is a
pretty neat little formula.


Finding the sum of n squares part 1

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.410
What I want to do in this video
is come up with an expression

00:00:04.410 --> 00:00:09.710
for finding the sum from i
equals 0 to n of i squared.

00:00:09.710 --> 00:00:11.510
So if I were to
expand this out, this

00:00:11.510 --> 00:00:17.250
is equal to 0 squared
plus 1 squared

00:00:17.250 --> 00:00:19.940
plus 2 squared plus 3 squared.

00:00:19.940 --> 00:00:22.786
And we're going to keep on
going all the way to n squared.

00:00:22.786 --> 00:00:24.660
So my goal is to find
some type of a function

00:00:24.660 --> 00:00:26.450
that you give me
the n and I will

00:00:26.450 --> 00:00:29.800
find the sum from 0
squared, 1 squared,

00:00:29.800 --> 00:00:31.300
2 squared, all the
way to n squared.

00:00:31.300 --> 00:00:32.966
And so you can imagine
that'd be useful,

00:00:32.966 --> 00:00:35.110
because this might be OK
if n is reasonably small.

00:00:35.110 --> 00:00:36.484
But if n is a big
number, this is

00:00:36.484 --> 00:00:38.350
going to take you forever to do.

00:00:38.350 --> 00:00:39.870
So let's first study this.

00:00:39.870 --> 00:00:42.640
Let's study what the input and
the output of this function

00:00:42.640 --> 00:00:43.680
needs to be.

00:00:43.680 --> 00:00:45.970
So the input is
going to be our n.

00:00:45.970 --> 00:00:50.044
So here we're starting with--
so n can go from 0 all the way--

00:00:50.044 --> 00:00:51.710
and we'll just try
up a bunch of values.

00:00:51.710 --> 00:00:53.600
So n could be 0.

00:00:53.600 --> 00:00:55.250
We could go from 0
all the way to 0.

00:00:55.250 --> 00:00:56.490
n could be 1.

00:00:56.490 --> 00:00:57.650
n could be 2.

00:00:57.650 --> 00:00:58.580
n could be 3.

00:00:58.580 --> 00:01:00.980
And we could just keep on
going on and on and on.

00:01:00.980 --> 00:01:02.630
But I'll just stop
there for now.

00:01:02.630 --> 00:01:04.549
Actually, let's just
go to 4, just for fun.

00:01:04.549 --> 00:01:06.090
And now, for each
of these, let's see

00:01:06.090 --> 00:01:07.840
what the output of our
function should be.

00:01:07.840 --> 00:01:10.020
The output of the function
should be this thing.

00:01:10.020 --> 00:01:16.560
It should be the sum from i
equals 0 to n of i squared.

00:01:16.560 --> 00:01:18.947
So when n is 0-- well, that's
just going to be 0 squared.

00:01:18.947 --> 00:01:20.280
We'd just stop right over there.

00:01:20.280 --> 00:01:21.780
So that's just 0.

00:01:21.780 --> 00:01:24.900
When n is 1, it's 0
squared plus 1 squared.

00:01:24.900 --> 00:01:26.410
So that is 1.

00:01:26.410 --> 00:01:29.930
When n is 2, it's 0 squared
plus 1 squared plus 2 squared.

00:01:29.930 --> 00:01:33.560
So that's 1 plus 4, which is 5.

00:01:33.560 --> 00:01:36.510
When n is 3, now we
go all the way to 3.

00:01:36.510 --> 00:01:40.440
So it's going to be 1 plus
4, which is 5, plus 9.

00:01:40.440 --> 00:01:43.970
So 5 plus 9 is 14.

00:01:43.970 --> 00:01:48.240
And then when n is 4, we're
going to add the 16, 4 squared,

00:01:48.240 --> 00:01:48.780
to this.

00:01:48.780 --> 00:01:50.030
So this gets us to 30.

00:01:50.030 --> 00:01:52.782
And of course, we could
keep going on and on and on.

00:01:52.782 --> 00:01:54.490
So let's study this
a little bit to think

00:01:54.490 --> 00:01:59.790
about what type of a function
that, for each of these inputs,

00:01:59.790 --> 00:02:02.554
might give us this
type of an output.

00:02:02.554 --> 00:02:04.970
So let's first look at the
difference between these terms.

00:02:04.970 --> 00:02:07.085
So the difference here is 1.

00:02:07.085 --> 00:02:09.167
The difference here is 4.

00:02:09.167 --> 00:02:10.000
And this is obvious.

00:02:10.000 --> 00:02:10.820
We added 1 here.

00:02:10.820 --> 00:02:12.550
We added 2 squared here.

00:02:12.550 --> 00:02:15.060
We added 3 squared, or 9, here.

00:02:15.060 --> 00:02:17.670
We added 4 squared, or 16, here.

00:02:17.670 --> 00:02:19.150
And the reason
why I'm doing this

00:02:19.150 --> 00:02:20.557
is if this was a
linear function,

00:02:20.557 --> 00:02:22.390
then the difference
between successive terms

00:02:22.390 --> 00:02:23.790
would be the same.

00:02:23.790 --> 00:02:25.840
Now, if this is a
quadratic function,

00:02:25.840 --> 00:02:28.730
then the differences between the
differences would be the same.

00:02:28.730 --> 00:02:30.580
Let's see if that's the case.

00:02:30.580 --> 00:02:31.870
So the difference here is 1.

00:02:31.870 --> 00:02:33.840
The difference here is 4.

00:02:33.840 --> 00:02:35.805
So the difference
between those is 3.

00:02:35.805 --> 00:02:39.500
The difference here is 5.

00:02:39.500 --> 00:02:41.970
The difference here is 7.

00:02:41.970 --> 00:02:44.840
So even the difference of the
differences is increasing.

00:02:44.840 --> 00:02:48.320
But if this is a cubic
function, then the differences

00:02:48.320 --> 00:02:50.960
of the difference of the
difference should be constant.

00:02:50.960 --> 00:02:52.660
So let's see if that's the case.

00:02:52.660 --> 00:02:54.160
And you'll appreciate
this even more

00:02:54.160 --> 00:02:55.430
when you start
learning calculus.

00:02:55.430 --> 00:02:55.971
So let's see.

00:02:55.971 --> 00:02:58.890
The difference
between 3 and 5 is 2.

00:02:58.890 --> 00:03:01.570
The difference
between 5 and 7 is 2.

00:03:01.570 --> 00:03:05.030
And so we keep having
a constant shift of 2.

00:03:05.030 --> 00:03:07.860
So the fact that the
difference of the difference

00:03:07.860 --> 00:03:10.990
of the difference
is fixed tells us

00:03:10.990 --> 00:03:12.880
that we should be
able to express this

00:03:12.880 --> 00:03:15.740
as some type of
a cubic function.

00:03:15.740 --> 00:03:18.160
So this we could
write as this should

00:03:18.160 --> 00:03:20.290
be equal to some
function in terms of n.

00:03:20.290 --> 00:03:26.890
And we could write it as An
to the third plus Bn squared

00:03:26.890 --> 00:03:31.330
plus C times n
plus D. And now we

00:03:31.330 --> 00:03:34.420
can just use what the inputs
are and the outputs are of these

00:03:34.420 --> 00:03:40.490
to solve for A, B, C, and D.
And I encourage you to do that.

00:03:40.490 --> 00:03:43.750
Well, let's first think
about when n is equal to 0.

00:03:43.750 --> 00:03:48.180
When n is equal to 0, this
function evaluates to D.

00:03:48.180 --> 00:03:53.030
So this function evaluates
to D. But that function

00:03:53.030 --> 00:03:56.650
needs to evaluate to
0, so D needs to be 0.

00:03:56.650 --> 00:03:59.410
So I'm just trying to
fix these letters here

00:03:59.410 --> 00:04:01.260
to get the right outputs.

00:04:01.260 --> 00:04:05.060
So when n is 0, this
expression evaluates to D.

00:04:05.060 --> 00:04:09.860
And it needs to evaluate to 0,
so D needs to be equal to 0.

00:04:09.860 --> 00:04:12.870
So D is equal to 0, or
we could just ignore it.

00:04:12.870 --> 00:04:14.450
So that helps us a little bit.

00:04:14.450 --> 00:04:16.269
We know from this
data point we're

00:04:16.269 --> 00:04:18.670
able to whittle it
down to it having

00:04:18.670 --> 00:04:21.120
this form right over here.

00:04:21.120 --> 00:04:24.570
And so now we can take
each of these inputs

00:04:24.570 --> 00:04:27.120
and figure out what their
corresponding output is.

00:04:27.120 --> 00:04:29.270
So let's do that--
I'll do that over here.

00:04:29.270 --> 00:04:31.980
So when n is 1,
this thing evaluates

00:04:31.980 --> 00:04:36.660
to-- let me do this in
a new color-- this thing

00:04:36.660 --> 00:04:42.725
evaluates to A times 1 to the
third power, which is just 1,

00:04:42.725 --> 00:04:46.800
plus B times 1 squared, which
is just 1, plus C times 1, which

00:04:46.800 --> 00:04:49.690
is just C. And this
needs to be equal to 1.

00:04:53.610 --> 00:05:01.420
Now, when n is 2, we have
A times n to the third.

00:05:01.420 --> 00:05:08.100
So that's 8A plus
2 squared is 4,

00:05:08.100 --> 00:05:14.715
plus 4B plus 2C needs
to be equal to 5.

00:05:17.360 --> 00:05:18.860
And I need to set
up three equations

00:05:18.860 --> 00:05:20.910
if I want to solve
for three unknowns.

00:05:20.910 --> 00:05:23.240
And so let's go to 3.

00:05:23.240 --> 00:05:31.240
So A times 3 to the third power,
so that's going to be 27A,

00:05:31.240 --> 00:05:44.180
plus 9B plus 3C is
going to be equal to 14.

00:05:44.180 --> 00:05:47.800
So I've set up three
equations in three unknowns.

00:05:47.800 --> 00:05:50.930
Now I just have to solve
these for A, B, and C.

00:05:50.930 --> 00:05:59.840
And I will have a generalized
formula for finding this sum

00:05:59.840 --> 00:06:05.470
right over here, the sum of
the first n numbers squared,

00:06:05.470 --> 00:06:06.760
I guess you could call it.

00:06:06.760 --> 00:06:09.160
So what I want to do now is
I'm going to stop this video.

00:06:09.160 --> 00:06:11.560
I encourage you to try to
solve the simultaneous equation

00:06:11.560 --> 00:06:12.060
on your own.

00:06:12.060 --> 00:06:15.760
In the next video, I'll
actually go and solve it.


Partial fraction decomposition to find sum of telescoping series

WEBVTT
Kind: captions
Language: en

00:00:00.551 --> 00:00:02.050
So what we're going
to attempt to do

00:00:02.050 --> 00:00:05.570
is evaluate this
sum right over here,

00:00:05.570 --> 00:00:08.109
evaluate what this
series is, negative 2

00:00:08.109 --> 00:00:11.660
over n plus 1 times n plus
2, starting at n equals 2,

00:00:11.660 --> 00:00:13.750
all the way to infinity.

00:00:13.750 --> 00:00:16.850
And if we wanted to see what
this looks like, it starts at n

00:00:16.850 --> 00:00:17.400
equals 2.

00:00:17.400 --> 00:00:20.430
So when n equals 2, this
is negative 2 over 2

00:00:20.430 --> 00:00:24.370
plus 1, which is 3, times
2 plus 2, which is 4.

00:00:24.370 --> 00:00:28.940
Then when n is equal to 3,
this is negative 2 over 3

00:00:28.940 --> 00:00:32.560
plus 1, which is 4, times
3 plus 2, which is 5.

00:00:32.560 --> 00:00:34.820
And it just keeps going
like that, negative 2

00:00:34.820 --> 00:00:37.670
over 5 times 6.

00:00:37.670 --> 00:00:40.620
And it just keeps
going on and on and on.

00:00:40.620 --> 00:00:43.880
And now, it looks pretty clear
that each successive term

00:00:43.880 --> 00:00:45.620
is getting smaller.

00:00:45.620 --> 00:00:47.680
And it's getting
smaller reasonably fast.

00:00:47.680 --> 00:00:51.814
So it's reasonable to assume
that even though you have

00:00:51.814 --> 00:00:53.480
an infinite number
of terms, it actually

00:00:53.480 --> 00:00:54.842
might give you a finite value.

00:00:54.842 --> 00:00:56.800
But it doesn't jump out
at me, at least the way

00:00:56.800 --> 00:00:58.470
that I've looked
at it right now,

00:00:58.470 --> 00:00:59.970
as to what this sum
would be, or how

00:00:59.970 --> 00:01:02.120
to actually figure out that sum.

00:01:02.120 --> 00:01:04.269
So what I want you to do
now is pause this video.

00:01:04.269 --> 00:01:07.630
And I'm going to give you a hint
about how to think about this.

00:01:07.630 --> 00:01:12.170
Try to dig up your memories
of partial fraction expansion,

00:01:12.170 --> 00:01:14.160
or partial fraction
decomposition,

00:01:14.160 --> 00:01:17.920
to turn this expression into
the sum of two fractions.

00:01:17.920 --> 00:01:22.280
And that might help us think
about what this sum is.

00:01:22.280 --> 00:01:24.337
So I'm assuming you've
given a go at it.

00:01:24.337 --> 00:01:25.920
So let's try to
manipulate this thing.

00:01:25.920 --> 00:01:28.840
And let's see if we can rewrite
this as a sum of two fractions.

00:01:28.840 --> 00:01:32.870
So this is negative
2 over-- and I'm

00:01:32.870 --> 00:01:35.990
going to do this in two
different colors-- n plus 1

00:01:35.990 --> 00:01:37.650
times n plus 2.

00:01:40.160 --> 00:01:42.870
And we remember from our
partial fraction expansion

00:01:42.870 --> 00:01:45.710
that we can rewrite this as
the sum of two fractions,

00:01:45.710 --> 00:01:54.846
as A over n plus 1
plus B over n plus 2.

00:01:54.846 --> 00:01:55.970
And why is this reasonable?

00:01:55.970 --> 00:01:57.090
Well, if you're
adding two fractions,

00:01:57.090 --> 00:01:58.620
you want to find a
common denominator, which

00:01:58.620 --> 00:02:00.650
would be a multiple of
the two denominators.

00:02:00.650 --> 00:02:03.300
This is clearly a multiple of
both of these denominators.

00:02:03.300 --> 00:02:05.660
And we learned in partial
fraction expansion

00:02:05.660 --> 00:02:08.919
that whatever we have up here,
especially because the degree

00:02:08.919 --> 00:02:13.020
here is lower than the degree
down here, whatever we have

00:02:13.020 --> 00:02:15.890
up here is going to be a degree
lower than what we have here.

00:02:15.890 --> 00:02:18.000
So this is a first-degree
term in terms of n,

00:02:18.000 --> 00:02:20.850
so these are going to be
constant terms up here.

00:02:20.850 --> 00:02:22.740
So let's figure out
what A and B are.

00:02:22.740 --> 00:02:25.580
So if we perform
the addition-- well,

00:02:25.580 --> 00:02:27.590
let's just rewrite
both of these with

00:02:27.590 --> 00:02:29.400
the same common denominator.

00:02:29.400 --> 00:02:34.190
So let's rewrite
A over n plus 1,

00:02:34.190 --> 00:02:37.720
but let's multiply the numerator
and denominator by n plus 2.

00:02:37.720 --> 00:02:41.570
So we multiply the numerator
by n plus 2 and the denominator

00:02:41.570 --> 00:02:42.100
by n plus 2.

00:02:42.100 --> 00:02:44.490
I haven't changed the value
of this first fraction.

00:02:44.490 --> 00:02:50.850
Similarly, let's do the same
thing with B over n plus 2.

00:02:50.850 --> 00:02:54.470
Multiply the numerator and the
denominator by n plus 1, so

00:02:54.470 --> 00:02:57.960
n plus 1 over n plus 1.

00:02:57.960 --> 00:03:01.276
Once again, I haven't change
the value of this fraction.

00:03:01.276 --> 00:03:03.400
But by doing this, I now
have a common denominator,

00:03:03.400 --> 00:03:04.570
and I can add.

00:03:04.570 --> 00:03:12.714
So this is going to be equal
to n plus 1 times n plus 2 is

00:03:12.714 --> 00:03:13.380
our denominator.

00:03:15.920 --> 00:03:19.690
And then our numerator--
let me expand it out.

00:03:19.690 --> 00:03:21.690
This is going to be,
if I distribute the A,

00:03:21.690 --> 00:03:25.290
it is An plus 2A.

00:03:25.290 --> 00:03:31.730
So let me write
that, An plus 2A.

00:03:31.730 --> 00:03:40.680
And then let's distribute
this B, plus Bn plus B.

00:03:40.680 --> 00:03:42.680
Now, what I want to do
is I want to rewrite this

00:03:42.680 --> 00:03:44.330
so I have all of the n terms.

00:03:44.330 --> 00:03:51.070
So for example, An plus Bn--
I could factor an n out.

00:03:51.070 --> 00:03:58.730
And I could rewrite that
as A plus B times n, those

00:03:58.730 --> 00:04:00.350
two terms right over there.

00:04:00.350 --> 00:04:03.895
And then these two
terms, the 2A plus B,

00:04:03.895 --> 00:04:09.080
I could just write it
like this, plus 2A plus B.

00:04:09.080 --> 00:04:17.550
And, of course, all of that is
over n plus 1 times n plus 2.

00:04:20.915 --> 00:04:24.020
So how do we solve for A and B?

00:04:24.020 --> 00:04:26.650
Well, the realization
is that this thing

00:04:26.650 --> 00:04:29.070
must be equal to negative 2.

00:04:29.070 --> 00:04:31.879
These two things must
be equal to each other.

00:04:31.879 --> 00:04:33.670
Remember, we're making
the claim that this,

00:04:33.670 --> 00:04:36.160
which is the same thing
as this, is equal to this.

00:04:36.160 --> 00:04:38.754
That's the whole reason
why we started doing this.

00:04:38.754 --> 00:04:40.420
So we're making the
claim that these two

00:04:40.420 --> 00:04:42.680
things are equivalent.

00:04:42.680 --> 00:04:44.470
We're making this claim.

00:04:44.470 --> 00:04:47.560
So everything in the numerator
must be equal to negative 2.

00:04:47.560 --> 00:04:48.710
So how do we do that?

00:04:48.710 --> 00:04:52.130
It looks like we have
two unknowns here.

00:04:52.130 --> 00:04:54.930
To figure out two unknowns, we
normally need two equations.

00:04:54.930 --> 00:04:56.990
Well, the realization
here is, look,

00:04:56.990 --> 00:05:00.030
we have an n term on
the left-hand side here.

00:05:00.030 --> 00:05:01.520
We have no n term here.

00:05:01.520 --> 00:05:03.950
So you literally could view
this, instead of just viewing

00:05:03.950 --> 00:05:05.366
this as negative
2, you could view

00:05:05.366 --> 00:05:10.960
this as negative 2 plus
0n, plus 0 times n.

00:05:10.960 --> 00:05:11.740
That's not "on."

00:05:11.740 --> 00:05:17.599
That's 0-- let me write
it this way-- 0 times n.

00:05:17.599 --> 00:05:19.140
So when you look at
it this way, it's

00:05:19.140 --> 00:05:22.430
clear that A plus B is
the coefficient on n.

00:05:22.430 --> 00:05:24.700
That must be equal to 0.

00:05:24.700 --> 00:05:28.340
A plus B must be equal to 0.

00:05:28.340 --> 00:05:30.810
And this is kind of
bread-and-butter partial

00:05:30.810 --> 00:05:32.180
fraction expansion.

00:05:32.180 --> 00:05:34.530
We have other videos on that
if you need to review that.

00:05:34.530 --> 00:05:43.010
And the constant part, 2A plus
B, is equal to negative 2.

00:05:46.100 --> 00:05:51.020
And so now we have two
equations in two unknowns.

00:05:51.020 --> 00:05:53.020
And we could solve it a
bunch of different ways.

00:05:53.020 --> 00:05:55.450
But one interesting way is
let's multiply the top equation

00:05:55.450 --> 00:05:56.930
by negative 1.

00:05:56.930 --> 00:06:01.090
So then this becomes negative
A minus B is equal to-- well,

00:06:01.090 --> 00:06:03.360
negative 1 times 0 is still 0.

00:06:03.360 --> 00:06:05.600
Now we can add these
two things together.

00:06:05.600 --> 00:06:11.110
And we are left with 2A minus
A is A, plus B minus B-- well,

00:06:11.110 --> 00:06:13.830
those cancel out.

00:06:13.830 --> 00:06:16.320
A is equal to negative 2.

00:06:16.320 --> 00:06:20.120
And if A is equal to
negative 2, A plus B is 0,

00:06:20.120 --> 00:06:21.520
B must be equal to 2.

00:06:24.090 --> 00:06:28.000
Negative 2 plus 2 is equal to 0.

00:06:28.000 --> 00:06:31.100
We solved for A. And then I
substituted it back up here.

00:06:31.100 --> 00:06:34.860
So now we can rewrite all
of this right over here.

00:06:34.860 --> 00:06:37.830
We can rewrite it as
the sum-- and actually,

00:06:37.830 --> 00:06:39.200
let me do a little bit instead.

00:06:39.200 --> 00:06:43.020
Let me just write it as
a finite sum as opposed

00:06:43.020 --> 00:06:44.200
to an infinite sum.

00:06:44.200 --> 00:06:47.450
And then we can just take the
limit as we go to infinity.

00:06:47.450 --> 00:06:49.190
So let me rewrite it like this.

00:06:49.190 --> 00:06:53.700
So this is the sum from n
equals 2-- instead to infinity,

00:06:53.700 --> 00:06:56.540
I'll just say to capital
N. And then later, we

00:06:56.540 --> 00:07:00.750
could take the limit as this
goes to infinity of-- well,

00:07:00.750 --> 00:07:03.850
instead of writing this, I can
write this right over here.

00:07:03.850 --> 00:07:06.370
So A is negative 2.

00:07:06.370 --> 00:07:11.110
So it's negative
2 over n plus 1.

00:07:11.110 --> 00:07:17.820
And then B is 2,
plus B over n plus 2.

00:07:17.820 --> 00:07:20.610
So once again, I've just
expressed this as a finite sum.

00:07:20.610 --> 00:07:23.210
Later, we can take the limit
as capital N approaches

00:07:23.210 --> 00:07:25.020
infinity to figure out
what this thing is.

00:07:25.020 --> 00:07:27.870
Oh, sorry, and B-- let
me not write B anymore.

00:07:27.870 --> 00:07:33.450
We now know that B
is 2 over n plus 2.

00:07:33.450 --> 00:07:37.850
Now, how does this actually
go about helping us?

00:07:37.850 --> 00:07:39.330
Well, let's do what
we did up here.

00:07:39.330 --> 00:07:42.260
Let's actually write out what
this is going to be equal to.

00:07:42.260 --> 00:07:46.900
This is going to be
equal to-- when n is 2,

00:07:46.900 --> 00:07:54.410
this is negative 2/3, so
it's negative 2/3, plus 2/4.

00:08:00.160 --> 00:08:02.810
So that's n equals-- let me
do it down here, because I'm

00:08:02.810 --> 00:08:04.400
about to run out of real estate.

00:08:04.400 --> 00:08:06.640
That is when n is equal to 2.

00:08:06.640 --> 00:08:10.120
Now, what about when
n is equal to 3?

00:08:10.120 --> 00:08:21.920
When n is equal to 3, this
is going to be negative 2/4

00:08:21.920 --> 00:08:22.610
plus 2/5.

00:08:28.770 --> 00:08:30.530
What about when n is equal to 4?

00:08:30.530 --> 00:08:33.890
I think you might see a pattern
that's starting to form.

00:08:33.890 --> 00:08:34.640
Let's do one more.

00:08:34.640 --> 00:08:42.090
When n is equal to
4, well, then, this

00:08:42.090 --> 00:08:46.710
is going to be
negative 2/5-- let

00:08:46.710 --> 00:08:53.170
me do that same blue color--
negative 2/5 plus 2/6.

00:08:57.640 --> 00:09:00.460
And we're just
going to keep going.

00:09:00.460 --> 00:09:02.520
Let me scroll down to
get some space-- we're

00:09:02.520 --> 00:09:05.190
going to keep going all the
way until the N-th term.

00:09:08.960 --> 00:09:13.680
So plus dot dot dot plus
our capital N-th term,

00:09:13.680 --> 00:09:24.110
which is going to be negative
2 over capital N plus 1 plus 2

00:09:24.110 --> 00:09:27.560
over capital N plus 2.

00:09:27.560 --> 00:09:29.310
So I think you might
see the pattern here.

00:09:29.310 --> 00:09:33.466
Notice, from our first when
n equals 2, we got the 2/4.

00:09:33.466 --> 00:09:35.590
But then when n equals 3,
you had the negative 2/4.

00:09:35.590 --> 00:09:37.450
That cancels with that.

00:09:37.450 --> 00:09:39.090
When n equals 3, you had 2/5.

00:09:39.090 --> 00:09:42.690
Then that cancels when n
equals 4 with the negative 2/5.

00:09:42.690 --> 00:09:47.170
So the second term cancels
with-- the second part,

00:09:47.170 --> 00:09:50.380
I guess, for each
n, for each index,

00:09:50.380 --> 00:09:53.040
cancels out with the first
part for the next index.

00:09:53.040 --> 00:09:55.420
And so that's just going to
keep happening all the way

00:09:55.420 --> 00:09:59.730
until n is equal to capital N.

00:09:59.730 --> 00:10:02.300
And so this is going to
cancel out with the one right

00:10:02.300 --> 00:10:03.110
before it.

00:10:03.110 --> 00:10:06.540
And all we're going
to be left with

00:10:06.540 --> 00:10:13.790
is this term and this
term right over here.

00:10:13.790 --> 00:10:15.950
So let's rewrite that.

00:10:15.950 --> 00:10:19.090
So we get-- let's
get more space here.

00:10:19.090 --> 00:10:26.290
This thing can be rewritten
as the sum from lowercase n

00:10:26.290 --> 00:10:30.850
equals 2 to capital
N of negative 2

00:10:30.850 --> 00:10:37.020
over n plus 1 plus
2 over n plus 2

00:10:37.020 --> 00:10:39.440
is equal to-- well, everything
else in the middle canceled

00:10:39.440 --> 00:10:39.940
out.

00:10:39.940 --> 00:10:43.740
We're just left
with negative 2/3

00:10:43.740 --> 00:10:50.460
plus 2 over capital N plus 2.

00:10:50.460 --> 00:10:53.110
So this was a huge
simplification right over here.

00:10:53.110 --> 00:10:57.000
And remember, our original sum
that we wanted to calculate,

00:10:57.000 --> 00:11:00.510
that just has a limit as
capital N goes to infinity.

00:11:00.510 --> 00:11:04.620
So let's just take the limit
as capital N goes to infinity.

00:11:04.620 --> 00:11:06.170
So let me write it this way.

00:11:06.170 --> 00:11:07.878
Well, actually, let
me write it this way.

00:11:07.878 --> 00:11:10.810
The limit-- so we can
write it this way.

00:11:10.810 --> 00:11:15.350
The limit as
capital N approaches

00:11:15.350 --> 00:11:20.030
infinity is going to be equal
to the limit as capital N

00:11:20.030 --> 00:11:22.239
approaches infinity
of-- well, we just

00:11:22.239 --> 00:11:23.280
figured out what this is.

00:11:23.280 --> 00:11:33.240
This is negative 2/3 plus
2 over capital N plus 2.

00:11:33.240 --> 00:11:36.180
Well, as n goes to
infinity, this negative 2/3

00:11:36.180 --> 00:11:37.600
doesn't get impacted at all.

00:11:37.600 --> 00:11:40.375
This term right over here, 2
over an ever larger number,

00:11:40.375 --> 00:11:42.000
over an infinitely
large number-- well,

00:11:42.000 --> 00:11:43.520
that's going to go to 0.

00:11:43.520 --> 00:11:47.990
And we're going to be
left with negative 2/3.

00:11:47.990 --> 00:11:48.750
And we're done.

00:11:48.750 --> 00:11:54.740
We were able to figure out the
sum of this infinite series.

00:11:54.740 --> 00:11:58.460
So this thing right over here
is equal to negative 2/3.

00:11:58.460 --> 00:12:01.320
And this type of series is
called a telescoping series--

00:12:01.320 --> 00:12:02.920
telescoping, I should say.

00:12:02.920 --> 00:12:04.380
This is a telescoping series.

00:12:09.270 --> 00:12:12.130
And a telescoping series
is a general term.

00:12:12.130 --> 00:12:14.500
So if you were to
take its partial sums,

00:12:14.500 --> 00:12:18.430
it has this pattern right over
here, where, in each term,

00:12:18.430 --> 00:12:20.020
you're starting to
cancel things out.

00:12:20.020 --> 00:12:23.270
So what you're left with
is just a fixed number

00:12:23.270 --> 00:12:25.520
of terms at the end.

00:12:25.520 --> 00:12:27.130
But either way,
this was a pretty--

00:12:27.130 --> 00:12:29.421
it's a little bit hairy, but
it was a pretty satisfying

00:12:29.421 --> 00:12:30.640
problem.


Explicitly defining a series

WEBVTT
Kind: captions
Language: en

00:00:00.342 --> 00:00:02.925
Let's say that we're told that
this sum right over here, where

00:00:02.925 --> 00:00:06.100
our index starts at 2 and we
go all the way to infinity,

00:00:06.100 --> 00:00:08.570
that this infinite
series is negative 8/5

00:00:08.570 --> 00:00:12.180
plus 16/7 minus 32/9
plus-- and we just

00:00:12.180 --> 00:00:14.130
keep going on and on forever.

00:00:14.130 --> 00:00:17.230
And so what I want to do
is to explicitly define

00:00:17.230 --> 00:00:18.979
what a sub n is here.

00:00:18.979 --> 00:00:20.520
So right now we just
say, hey, if you

00:00:20.520 --> 00:00:24.000
take the sum of a sub n
from n equals 2 to infinity,

00:00:24.000 --> 00:00:26.177
it turns out you get
this sum right over here.

00:00:26.177 --> 00:00:28.510
But let's think about what a
sub n-- how we can actually

00:00:28.510 --> 00:00:30.872
define it in terms of n.

00:00:30.872 --> 00:00:33.330
And I encourage you to pause
the video right now and try it

00:00:33.330 --> 00:00:34.580
on your own.

00:00:34.580 --> 00:00:36.810
So the first thing
that you might realize

00:00:36.810 --> 00:00:40.390
is, well, this is the number
that we're going to get.

00:00:40.390 --> 00:00:41.970
Let me write it this way.

00:00:41.970 --> 00:00:47.920
a sub 2 is equal
to negative 8/5.

00:00:47.920 --> 00:00:52.995
a sub 3 is equal to 16/7.

00:00:56.090 --> 00:00:59.540
a sub 4 is equal
to negative 32/9.

00:01:02.280 --> 00:01:04.940
And I'm just giving the sign
to the number in the numerator.

00:01:04.940 --> 00:01:07.760
Negative 8/5 is the same
thing as negative 8 over 5.

00:01:07.760 --> 00:01:10.720
Let me make that a
little bit clearer.

00:01:10.720 --> 00:01:13.000
So I'll make that a
little bit clearer.

00:01:13.000 --> 00:01:15.027
So this is negative 8/5.

00:01:15.027 --> 00:01:16.402
Obviously, this
is positive, so I

00:01:16.402 --> 00:01:18.276
don't have to really
worry about it too much.

00:01:18.276 --> 00:01:20.560
And then here, I'm just
saying negative 32/9,

00:01:20.560 --> 00:01:25.850
so it's the same thing
as negative 32 over 9.

00:01:25.850 --> 00:01:27.910
So let's see if
we can first find

00:01:27.910 --> 00:01:31.730
a pattern in the numerator.

00:01:31.730 --> 00:01:35.840
So when we go from negative
8 to 16, what's happening?

00:01:35.840 --> 00:01:39.330
Well, we're multiplying
by negative 2.

00:01:41.860 --> 00:01:44.120
Now, to go from
16 to negative 32,

00:01:44.120 --> 00:01:49.020
we're multiplying
by negative 2 again.

00:01:49.020 --> 00:01:50.640
So you might say,
OK, well, whatever

00:01:50.640 --> 00:01:54.037
we have in the numerator must
be a power of negative 2.

00:01:54.037 --> 00:01:55.870
And, all right, if you
say, well, maybe this

00:01:55.870 --> 00:01:57.710
is negative 2 squared, well,
you know that negative 8

00:01:57.710 --> 00:01:58.850
isn't negative 2 squared.

00:01:58.850 --> 00:02:01.610
Negative 2 squared is
equal to positive 4.

00:02:01.610 --> 00:02:04.150
Negative 8-- this
right over here.

00:02:04.150 --> 00:02:10.610
Negative 8, that is equal to
negative 2 to the third power.

00:02:10.610 --> 00:02:16.470
16 is equal to negative
2 to the fourth power.

00:02:16.470 --> 00:02:26.400
Negative 32 is equal to
negative 2 to the fifth power.

00:02:26.400 --> 00:02:28.700
So notice, our exponent
on the negative 2

00:02:28.700 --> 00:02:32.960
is always going to be
one more than our index.

00:02:32.960 --> 00:02:34.400
Our index is 2,
our exponent is 3.

00:02:34.400 --> 00:02:36.670
Our index is 3,
our exponent is 4.

00:02:36.670 --> 00:02:39.730
Our index is 4,
our exponent is 5.

00:02:39.730 --> 00:02:42.930
So that gives a sense that at
least the numerator is going

00:02:42.930 --> 00:02:45.840
to be-- whatever our index
is, it's going to be-- so let

00:02:45.840 --> 00:02:46.790
me write this down.

00:02:46.790 --> 00:02:53.070
So a sub n is equal
to-- well, it's

00:02:53.070 --> 00:02:56.170
going to be negative 2 to
whatever index we're at,

00:02:56.170 --> 00:02:59.350
to that index plus 1 power.

00:02:59.350 --> 00:03:02.580
So that's a reasonable way
to think about our numerator.

00:03:02.580 --> 00:03:04.790
Now let's think about
our denominators.

00:03:04.790 --> 00:03:10.760
So over-- So we go from 5,
so when n is 2, we're at 5.

00:03:10.760 --> 00:03:13.340
When n is 3, we're at 7.

00:03:13.340 --> 00:03:16.500
When n is 4, we're at 9.

00:03:16.500 --> 00:03:19.785
So notice, 5 is
2 times 2 plus 1.

00:03:27.460 --> 00:03:32.400
This right over here
is 2 times 3 plus 1.

00:03:32.400 --> 00:03:37.685
This right over here
is 2 times 4 plus 1.

00:03:37.685 --> 00:03:39.310
And you should just
kind of play around

00:03:39.310 --> 00:03:40.810
with different
patterns in your head

00:03:40.810 --> 00:03:42.680
until you say, hey,
well, look, this

00:03:42.680 --> 00:03:44.870
is increasing by 2 every time.

00:03:44.870 --> 00:03:48.670
Notice, this increases
by 2 every time.

00:03:48.670 --> 00:03:51.680
But these aren't
exactly multiples of 2.

00:03:51.680 --> 00:03:55.045
These seem to be off by one more
than the multiples of 2, which

00:03:55.045 --> 00:03:56.420
is a good sign
that this is going

00:03:56.420 --> 00:03:59.340
to be 2 times our index plus 1.

00:03:59.340 --> 00:04:05.665
So we could write this as
2 times our index plus 1.

00:04:05.665 --> 00:04:06.290
And we're done.

00:04:06.290 --> 00:04:07.420
That's what a sub n is.

00:04:07.420 --> 00:04:10.780
And if we wanted to write
this series in sigma notation,

00:04:10.780 --> 00:04:12.630
we would write this
as the sum from n

00:04:12.630 --> 00:04:16.880
equals 2 to infinity
of negative 2

00:04:16.880 --> 00:04:21.269
to the n plus 1
power over 2n plus 1.

00:04:21.269 --> 00:04:24.790
And that would equal this
series right over here.


Writing a series in sigma notation

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:06.250
So I have the series-- negative
5/3 plus 25 over 6 minus 125

00:00:06.250 --> 00:00:09.950
over 9 plus-- and it just keeps
going on and on and on forever.

00:00:09.950 --> 00:00:12.240
So this right over
here is an infinite sum

00:00:12.240 --> 00:00:15.360
or an infinite series, and
what I want you to do right now

00:00:15.360 --> 00:00:18.320
is to pause this video
and try to express

00:00:18.320 --> 00:00:23.490
this infinite series
using sigma notation.

00:00:23.490 --> 00:00:26.390
So I'm assuming you've given
a go at it, so let's just

00:00:26.390 --> 00:00:29.670
look at each term of the series
and let's see if we can express

00:00:29.670 --> 00:00:33.340
it with kind of an
ever-increasing index.

00:00:33.340 --> 00:00:35.630
So the first thing that
might jump out at you

00:00:35.630 --> 00:00:40.350
is this oscillating sign that's
happening right over here.

00:00:40.350 --> 00:00:42.400
And whenever we see
an oscillating sign,

00:00:42.400 --> 00:00:44.358
that's a pretty good idea
that we could kind of

00:00:44.358 --> 00:00:48.360
view this as negative 1 to the
n-th, where n is our index.

00:00:48.360 --> 00:00:51.020
So for example, that
right over there

00:00:51.020 --> 00:00:54.580
is negative 1 to
the first power.

00:00:54.580 --> 00:01:01.440
That is-- this right
over here is negative 1

00:01:01.440 --> 00:01:03.080
to the second power.

00:01:03.080 --> 00:01:08.190
That right over there is
negative 1 to the third power.

00:01:08.190 --> 00:01:10.610
So it looks like the
sign is being defined

00:01:10.610 --> 00:01:13.120
by raising negative
1 to the index.

00:01:13.120 --> 00:01:16.340
Now let's look at the
other parts of these terms

00:01:16.340 --> 00:01:17.640
right over here.

00:01:17.640 --> 00:01:22.290
So we have 5, then we
have 25, the we have 125,

00:01:22.290 --> 00:01:24.420
so these are the powers of 5.

00:01:24.420 --> 00:01:27.250
So this right over here
is 5 to the first power,

00:01:27.250 --> 00:01:29.980
this right over here is
5 to the second power,

00:01:29.980 --> 00:01:33.280
this right over here is
5 to the third power.

00:01:33.280 --> 00:01:35.940
So this part, we're
raising 5 to our index.

00:01:35.940 --> 00:01:40.370
Notice, 1, 1, 2, 2, 3, 3.

00:01:40.370 --> 00:01:42.350
And then finally,
let's look at this.

00:01:42.350 --> 00:01:46.260
We have 3, 6, and 9.

00:01:46.260 --> 00:01:49.730
So this literally-- if our index
here is 1, this is 3 times 1.

00:01:49.730 --> 00:01:52.160
If our index here is
2, this is 3 times 2.

00:01:52.160 --> 00:01:55.110
If our index here is
3, this is 3 times 3.

00:01:55.110 --> 00:02:01.250
So this is 3 times 1,
that is 3 times 2--

00:02:01.250 --> 00:02:03.470
let me write it this
way-- 3 times 2,

00:02:03.470 --> 00:02:06.940
that right over
there is 3 times 3.

00:02:06.940 --> 00:02:09.630
So this sets us up
pretty well to write

00:02:09.630 --> 00:02:11.160
this in sigma notation.

00:02:11.160 --> 00:02:14.440
So let's write it over here
just so we can compare.

00:02:14.440 --> 00:02:20.710
So let me give myself some
real estate to work with.

00:02:20.710 --> 00:02:25.140
So we could write
this as the sum--

00:02:25.140 --> 00:02:28.535
I'll do it in yellow-- as the
sum, so this is our sigma.

00:02:31.670 --> 00:02:37.950
We can start our index
n at 1, from n equals 1,

00:02:37.950 --> 00:02:39.950
and we're going to keep
going on and on forever.

00:02:39.950 --> 00:02:42.190
We just keep going
on and on forever.

00:02:42.190 --> 00:02:50.120
And so it's negative 1
to the n-th power times 5

00:02:50.120 --> 00:02:59.250
to the n-th over-- notice
5 to the n-th-- over 3n

00:02:59.250 --> 00:03:00.700
is going to be equal to this.

00:03:00.700 --> 00:03:02.270
And you can verify
when n equals 1,

00:03:02.270 --> 00:03:03.771
it's negative 1
to the nth power--

00:03:03.771 --> 00:03:05.770
I'm sorry-- negative 1
to the first power, which

00:03:05.770 --> 00:03:08.200
is negative 1 times 5 to
the first power, which

00:03:08.200 --> 00:03:10.170
is 5 over 3 times 1.

00:03:10.170 --> 00:03:12.335
And we can do that for
each successive term.

00:03:12.335 --> 00:03:15.660
And so we're all done.


Worked example: sequence convergence/divergence

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.960
So we've explicitly defined
four different sequences here.

00:00:03.960 --> 00:00:05.550
And what I want
you to think about

00:00:05.550 --> 00:00:09.970
is whether these sequences
converge or diverge.

00:00:09.970 --> 00:00:11.700
And remember,
converge just means,

00:00:11.700 --> 00:00:13.705
as n gets larger and
larger and larger,

00:00:13.705 --> 00:00:17.620
that the value of our sequence
is approaching some value.

00:00:17.620 --> 00:00:21.254
And diverge means that it's
not approaching some value.

00:00:21.254 --> 00:00:22.170
So let's look at this.

00:00:22.170 --> 00:00:23.794
And I encourage you
to pause this video

00:00:23.794 --> 00:00:26.510
and try this on your own
before I'm about to explain it.

00:00:26.510 --> 00:00:29.110
So let's look at this first
sequence right over here.

00:00:29.110 --> 00:00:32.729
So the numerator n plus 8 times
n plus 1, the denominator n

00:00:32.729 --> 00:00:35.070
times n minus 10.

00:00:35.070 --> 00:00:37.530
So one way to think about
what's happening as n

00:00:37.530 --> 00:00:40.074
gets larger and larger is look
at the degree of the numerator

00:00:40.074 --> 00:00:41.490
and the degree of
the denominator.

00:00:41.490 --> 00:00:43.906
And we care about the degree
because we want to see, look,

00:00:43.906 --> 00:00:46.260
is the numerator growing
faster than the denominator?

00:00:46.260 --> 00:00:48.604
In which case this thing
is going to go to infinity

00:00:48.604 --> 00:00:50.020
and this thing's
going to diverge.

00:00:50.020 --> 00:00:52.870
Or is maybe the denominator
growing faster, in which case

00:00:52.870 --> 00:00:55.180
this might converge to 0?

00:00:55.180 --> 00:00:57.040
Or maybe they're growing
at the same level,

00:00:57.040 --> 00:00:59.699
and maybe it'll converge
to a different number.

00:00:59.699 --> 00:01:01.990
So let's multiply out the
numerator and the denominator

00:01:01.990 --> 00:01:03.270
and figure that out.

00:01:03.270 --> 00:01:06.330
So n times n is n squared.

00:01:06.330 --> 00:01:11.900
n times 1 is 1n, plus 8n is 9n.

00:01:11.900 --> 00:01:14.180
And then 8 times 1 is 8.

00:01:14.180 --> 00:01:17.020
So the numerator is n
squared plus 9n plus 8.

00:01:17.020 --> 00:01:22.960
The denominator is
n squared minus 10n.

00:01:22.960 --> 00:01:24.400
And one way to
think about it is n

00:01:24.400 --> 00:01:27.290
gets really, really, really,
really, really large,

00:01:27.290 --> 00:01:29.930
what dominates in the
numerator-- this term is going

00:01:29.930 --> 00:01:31.410
to represent most of the value.

00:01:31.410 --> 00:01:33.660
And this term is going to
represent most of the value,

00:01:33.660 --> 00:01:34.210
as well.

00:01:34.210 --> 00:01:36.687
These other terms
aren't going to grow.

00:01:36.687 --> 00:01:38.270
Obviously, this 8
doesn't grow at all.

00:01:38.270 --> 00:01:41.200
But the n terms aren't going
to grow anywhere near as fast

00:01:41.200 --> 00:01:44.670
as the n squared terms,
especially for large n's.

00:01:44.670 --> 00:01:47.080
So for very, very
large n's, this

00:01:47.080 --> 00:01:48.470
is really going
to be approaching

00:01:48.470 --> 00:01:51.600
n squared over n squared, or 1.

00:01:51.600 --> 00:01:54.570
So it's reasonable to
say that this converges.

00:01:54.570 --> 00:01:56.735
So this one converges.

00:01:56.735 --> 00:01:58.860
And once again, I'm not
vigorously proving it here.

00:01:58.860 --> 00:02:00.990
Or I should say
I'm not rigorously

00:02:00.990 --> 00:02:02.210
proving it over here.

00:02:02.210 --> 00:02:05.240
But the giveaway is that
we have the same degree

00:02:05.240 --> 00:02:07.950
in the numerator
and the denominator.

00:02:07.950 --> 00:02:09.990
So now let's look at
this one right over here.

00:02:09.990 --> 00:02:13.150
So here in the numerator
I have e to the n power.

00:02:13.150 --> 00:02:15.980
And here I have e times n.

00:02:15.980 --> 00:02:21.030
So this grows much faster.

00:02:21.030 --> 00:02:23.130
I mean, this is
e to the n power.

00:02:23.130 --> 00:02:25.140
Imagine if when you
have this as 100,

00:02:25.140 --> 00:02:28.110
e to the 100th power is a
ginormous number. e times 100--

00:02:28.110 --> 00:02:29.430
that's just 100e.

00:02:29.430 --> 00:02:36.340
Grows much faster than
this right over here.

00:02:36.340 --> 00:02:38.170
So this thing is just
going to balloon.

00:02:38.170 --> 00:02:39.586
This is going to go to infinity.

00:02:39.586 --> 00:02:40.835
So we could say this diverges.

00:02:44.200 --> 00:02:46.140
Now let's look at this
one right over here.

00:02:46.140 --> 00:02:48.420
Well, we have a
higher degree term.

00:02:48.420 --> 00:02:50.090
We have a higher
degree in the numerator

00:02:50.090 --> 00:02:51.520
than we have in the denominator.

00:02:51.520 --> 00:02:54.510
n squared, obviously, is going
to grow much faster than n.

00:02:54.510 --> 00:02:58.130
So for the same reason
as the b sub n sequence,

00:02:58.130 --> 00:02:59.750
this thing is going to diverge.

00:02:59.750 --> 00:03:01.680
The numerator is going
to grow much faster

00:03:01.680 --> 00:03:02.654
than the denominator.

00:03:02.654 --> 00:03:04.570
Or another way to think
about it, the limit as

00:03:04.570 --> 00:03:06.970
n approaches infinity
is going to be infinity.

00:03:06.970 --> 00:03:09.230
This thing's going
to go to infinity.

00:03:09.230 --> 00:03:11.510
Now let's think about
this right over here.

00:03:11.510 --> 00:03:14.562
So as we increase
n-- so we could even

00:03:14.562 --> 00:03:16.270
think about what the
sequence looks like.

00:03:16.270 --> 00:03:19.180
When n is 0, negative
1 to the 0 is 1.

00:03:19.180 --> 00:03:21.350
When n is 1, it's
going to be negative 1.

00:03:21.350 --> 00:03:23.335
When n is 2, it's going to be 1.

00:03:23.335 --> 00:03:24.960
And so this thing is
just going to keep

00:03:24.960 --> 00:03:27.710
oscillating between
negative 1 and 1.

00:03:27.710 --> 00:03:29.350
So it's not unbounded.

00:03:29.350 --> 00:03:31.700
It's not going to go to
infinity or negative infinity

00:03:31.700 --> 00:03:33.160
or something like that.

00:03:33.160 --> 00:03:35.300
But it just oscillates
between these two values.

00:03:35.300 --> 00:03:39.010
So it doesn't converge
to one particular value.

00:03:39.010 --> 00:03:41.290
So even though this one
isn't unbounded-- it doesn't

00:03:41.290 --> 00:03:43.870
go to infinity-- this
one still diverges.

00:03:43.870 --> 00:03:46.697
It doesn't go to one value.

00:03:46.697 --> 00:03:47.780
So let me write that down.

00:03:47.780 --> 00:03:50.530
This one diverges.


Sn1 and Sn2 solvents effects

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.310
So right here I have
two potential solvents

00:00:03.310 --> 00:00:06.450
that you might see some type
of reaction take place in.

00:00:06.450 --> 00:00:08.760
The one on the right is a
very typical one, probably

00:00:08.760 --> 00:00:10.420
the most common of all solvents.

00:00:10.420 --> 00:00:11.670
This is water.

00:00:11.670 --> 00:00:14.990
And this right over
here is diethyl ether.

00:00:14.990 --> 00:00:16.720
And we'll talk
about naming things

00:00:16.720 --> 00:00:18.470
like this in the
future dates, but you

00:00:18.470 --> 00:00:20.340
can see what's going
on right over here.

00:00:20.340 --> 00:00:22.230
You have two ethyl
groups attached

00:00:22.230 --> 00:00:25.790
to this oxygen
between diethyl ether.

00:00:25.790 --> 00:00:27.670
But the important point
here is to realize

00:00:27.670 --> 00:00:29.679
where are these two
solvents similar?

00:00:29.679 --> 00:00:31.220
Where are these two
molecules similar

00:00:31.220 --> 00:00:32.549
and where are they different?

00:00:32.549 --> 00:00:35.600
The places that they are
similar is in both cases oxygen,

00:00:35.600 --> 00:00:37.580
which is one of the most
electronegative of all

00:00:37.580 --> 00:00:39.550
of the atoms, that
in both cases,

00:00:39.550 --> 00:00:42.400
it's bonded to things that
are less electronegative.

00:00:42.400 --> 00:00:47.200
And so that allows oxygen to
hog the electrons more often.

00:00:47.200 --> 00:00:49.170
It has a higher
electron affinity.

00:00:49.170 --> 00:00:53.220
And so you'll have a
partial negative charge

00:00:53.220 --> 00:00:57.710
for both of these molecules
on the oxygen side.

00:00:57.710 --> 00:01:02.930
And you'd have a
partial positive side

00:01:02.930 --> 00:01:05.060
from the thing that's
getting to spend

00:01:05.060 --> 00:01:10.510
less time with the electrons.

00:01:10.510 --> 00:01:12.850
So both of these we could
call them polar solvents.

00:01:17.800 --> 00:01:20.500
And a general idea
throughout chemistry

00:01:20.500 --> 00:01:24.100
is that polar solvents are
better at dissolving ions.

00:01:24.100 --> 00:01:26.876
Because a polar
solvent you have kind

00:01:26.876 --> 00:01:29.250
of positive negative charges,
things are flipping around,

00:01:29.250 --> 00:01:34.580
they're partially attracted
to the other positive ends

00:01:34.580 --> 00:01:35.800
of the other molecule.

00:01:35.800 --> 00:01:38.680
You could imagine that
it's very easy for an ion,

00:01:38.680 --> 00:01:41.790
for something that is positively
or negatively charged here,

00:01:41.790 --> 00:01:44.840
to kind of get mixed
in to the fray.

00:01:44.840 --> 00:01:46.480
Now the difference
between these two

00:01:46.480 --> 00:01:49.320
is obviously what the
oxygen is attached to.

00:01:49.320 --> 00:01:52.090
Here the oxygen is
attached to some carbons.

00:01:52.090 --> 00:01:54.730
Here it is attached
to hydrogens.

00:01:54.730 --> 00:01:58.212
And oxygen is far
more electronegative

00:01:58.212 --> 00:01:59.170
than the hydrogens are.

00:01:59.170 --> 00:02:01.044
It's a much bigger
differential than the case

00:02:01.044 --> 00:02:03.590
between oxygen and carbon.

00:02:03.590 --> 00:02:06.172
So in this situation
you will see scenarios

00:02:06.172 --> 00:02:07.880
where it is possible
where in a mixture--

00:02:07.880 --> 00:02:09.929
remember these things
get really, really messy.

00:02:09.929 --> 00:02:11.720
There's a lot of just
random bumping that's

00:02:11.720 --> 00:02:15.540
going on over here-- that oxygen
is so electronegative that it

00:02:15.540 --> 00:02:19.690
can often times take
this pair that's

00:02:19.690 --> 00:02:21.580
in this bond right over there.

00:02:21.580 --> 00:02:26.920
And if it did that you would be
left with the oxygen now having

00:02:26.920 --> 00:02:28.020
a negative charge.

00:02:30.379 --> 00:02:31.920
Before this-- let's
see if I can draw

00:02:31.920 --> 00:02:33.128
all of its valence electrons.

00:02:33.128 --> 00:02:36.730
It was one, two,
three, four, five, six.

00:02:36.730 --> 00:02:39.549
So now it's going to
have one, two, three,

00:02:39.549 --> 00:02:41.840
four-- actually maybe I can
draw them a little bit more

00:02:41.840 --> 00:02:42.810
carefully.

00:02:42.810 --> 00:02:47.960
So you still have, you could say
half of this bond to hydrogen.

00:02:47.960 --> 00:02:49.840
Now you have these two pairs.

00:02:49.840 --> 00:02:52.010
So one, two, three, four.

00:02:52.010 --> 00:02:53.800
And now it gains another pair.

00:02:53.800 --> 00:02:55.200
It got both of these electrons.

00:02:55.200 --> 00:02:57.200
It was already kind of
sharing half of this bond

00:02:57.200 --> 00:02:59.960
but now it has the whole bond.

00:02:59.960 --> 00:03:01.420
It has a negative charge.

00:03:01.420 --> 00:03:02.720
And then you have the hydrogen.

00:03:02.720 --> 00:03:04.810
Well if you strip
it of its electron,

00:03:04.810 --> 00:03:08.910
hydrogen is just a
proton floating around.

00:03:08.910 --> 00:03:10.660
So let me draw that.

00:03:10.660 --> 00:03:17.850
So this hydrogen is now just
a proton floating around.

00:03:17.850 --> 00:03:19.740
So when you have water,
water is actually

00:03:19.740 --> 00:03:21.209
called a protic solvent.

00:03:21.209 --> 00:03:22.750
And then maybe
another water molecule

00:03:22.750 --> 00:03:24.910
might pick this up
and take it someplace

00:03:24.910 --> 00:03:27.440
forming a hydronium cation.

00:03:27.440 --> 00:03:30.430
But this right over
here, water is protic.

00:03:30.430 --> 00:03:33.260
Which means that you might have
these protons floating around,

00:03:33.260 --> 00:03:34.690
these hydrogen protons.

00:03:34.690 --> 00:03:36.960
While diethyl ether
is not protic.

00:03:36.960 --> 00:03:39.540
Even in the case that
somehow one of the bonds--

00:03:39.540 --> 00:03:42.360
you don't just have
little hydrogen protons

00:03:42.360 --> 00:03:43.667
floating around in this.

00:03:43.667 --> 00:03:45.250
And so what I want
to now think about,

00:03:45.250 --> 00:03:48.320
now that we know about polar
solvents, why they're polar.

00:03:48.320 --> 00:03:52.130
But we also know examples of
protic and nonprotic solvents.

00:03:52.130 --> 00:03:55.830
Let's think about
which might be better

00:03:55.830 --> 00:03:57.950
when we're talking about
the types of reactions

00:03:57.950 --> 00:03:59.220
we're studying.

00:03:59.220 --> 00:04:01.375
So right over here you
might recognize-- in fact,

00:04:01.375 --> 00:04:02.250
it's a good exercise.

00:04:02.250 --> 00:04:04.890
Pause this video and
tell me these top three

00:04:04.890 --> 00:04:06.750
steps, what type of
reaction is this,

00:04:06.750 --> 00:04:10.150
and these bottom two what
type of reaction is this.

00:04:10.150 --> 00:04:14.730
Well, in this top one we see
that this leaving group leaves

00:04:14.730 --> 00:04:18.010
first, and that this is
the rate determining step.

00:04:18.010 --> 00:04:19.924
And then a nucleophile
comes in here

00:04:19.924 --> 00:04:22.600
and attacks the
carbocation and attaches.

00:04:22.600 --> 00:04:25.050
So this is a
substitution reaction,

00:04:25.050 --> 00:04:28.050
but the rate that with a
nucleophile, but the rate

00:04:28.050 --> 00:04:30.410
determining step involves
only one of the reactants.

00:04:30.410 --> 00:04:32.200
The leaving group's
got to leave.

00:04:32.200 --> 00:04:37.176
So this right over here is an
SN1 reaction, only involving

00:04:37.176 --> 00:04:38.050
one of the reactants.

00:04:38.050 --> 00:04:41.080
The leaving group leaving is
the rate determining step.

00:04:41.080 --> 00:04:44.550
This, on the other hand,
both the nucleophile attacks

00:04:44.550 --> 00:04:47.350
and the leaving group
leaves in one step.

00:04:47.350 --> 00:04:49.850
So this is a substitution
involving a nucleophile,

00:04:49.850 --> 00:04:53.490
but both reactants are involved
in the rate determining step.

00:04:53.490 --> 00:04:56.870
Now since both SN1
and SN2 reactions

00:04:56.870 --> 00:05:00.430
need these ions
floating around you

00:05:00.430 --> 00:05:05.400
can imagine that they both
benefit from polar solvents.

00:05:05.400 --> 00:05:07.310
So polar solvents
are good for both.

00:05:15.564 --> 00:05:16.980
Now let's think
about whether it's

00:05:16.980 --> 00:05:19.860
good to have protic or
nonprotic environments

00:05:19.860 --> 00:05:24.210
for either one of
these reactions.

00:05:24.210 --> 00:05:25.960
Well to think about
that a little bit more

00:05:25.960 --> 00:05:27.751
let's think about the
types of nucleophiles

00:05:27.751 --> 00:05:30.160
and the types of molecules
we're dealing with.

00:05:30.160 --> 00:05:32.265
SN2 reactions, in order
for this to happen you

00:05:32.265 --> 00:05:34.140
have to have a reasonably
strong nucleophile.

00:05:42.680 --> 00:05:45.380
And the SN1 reaction,
well you already

00:05:45.380 --> 00:05:47.290
have the carbocation
set up here,

00:05:47.290 --> 00:05:49.520
the leaving group leaving
was actually the hard part.

00:05:49.520 --> 00:05:51.725
So this could even work
with a weak nucleophile.

00:05:56.490 --> 00:05:59.270
The real thing you have to do
is to get this leaving group

00:05:59.270 --> 00:05:59.986
to leave.

00:05:59.986 --> 00:06:01.610
And also if we want
to remind ourselves

00:06:01.610 --> 00:06:03.609
of some of the other kind
of the stereochemistry

00:06:03.609 --> 00:06:07.320
of the reactions,
in an SN1 reaction,

00:06:07.320 --> 00:06:10.130
it's actually nice to
have a tertiary carbon.

00:06:10.130 --> 00:06:13.470
If these are also carbons, these
little circles I drew here,

00:06:13.470 --> 00:06:15.720
that helps stabilize
the carbocation.

00:06:15.720 --> 00:06:18.920
So if this was a tertiary
carbon this would favor SN1.

00:06:18.920 --> 00:06:21.470
We also know that tertiary
carbon would not be good.

00:06:21.470 --> 00:06:24.390
If this is a tertiary carbon
that would not be good for SN2.

00:06:24.390 --> 00:06:27.580
Because if these were
carbon chains of some

00:06:27.580 --> 00:06:30.100
form-- these three,
this purple, this blue,

00:06:30.100 --> 00:06:31.834
and this orange
thing-- then they

00:06:31.834 --> 00:06:33.000
would have steric hindrance.

00:06:33.000 --> 00:06:35.300
The nucleophile would
have trouble getting in.

00:06:35.300 --> 00:06:36.920
So this would be
more likely to happen

00:06:36.920 --> 00:06:40.300
if you're dealing with a methyl
carbon or a primary carbon,

00:06:40.300 --> 00:06:42.640
maybe a secondary carbon.

00:06:42.640 --> 00:06:44.830
So here the nucleophile
is reasonably strong.

00:06:44.830 --> 00:06:47.760
Here it can even work
with a weak nucleophile.

00:06:47.760 --> 00:06:50.220
Polar solvent in either case.

00:06:50.220 --> 00:06:54.130
And the SN1 the key is
stabilizing the leaving group,

00:06:54.130 --> 00:06:56.810
making it OK for the
leaving group to leave.

00:06:56.810 --> 00:06:58.940
So what's going to help
a leaving group to leave?

00:06:58.940 --> 00:07:00.523
Especially when a
leaving group leaves

00:07:00.523 --> 00:07:02.590
it typically has
a negative charge.

00:07:02.590 --> 00:07:05.370
Well if there were a bunch
of protons flying around that

00:07:05.370 --> 00:07:07.790
could help stabilize
this negative charge,

00:07:07.790 --> 00:07:08.590
that would help.

00:07:08.590 --> 00:07:11.760
In fact, this could then form a
bond with one of those protons.

00:07:11.760 --> 00:07:14.800
It could donate electrons
to one of those protons.

00:07:14.800 --> 00:07:22.220
And so a protic environment
is good for the SN1.

00:07:22.220 --> 00:07:26.006
While on the SN2 since you
have these strong nucleophiles,

00:07:26.006 --> 00:07:28.130
if you had protons flying
around, instead of trying

00:07:28.130 --> 00:07:30.000
to attack the substrate
right over here,

00:07:30.000 --> 00:07:32.160
the nucleophile would
attack the protons.

00:07:32.160 --> 00:07:34.980
So you don't want that, then
this thing won't happen.

00:07:34.980 --> 00:07:42.911
So you want nonprotic
for the SN2.

00:07:42.911 --> 00:07:44.410
So hopefully that
gives a little bit

00:07:44.410 --> 00:07:46.520
of the basics of the
solvents and also

00:07:46.520 --> 00:07:50.530
allows us to compare these two
reactions in interesting ways.


Sn2 stereochemistry

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:02.710
What I have depicted
right over here

00:00:02.710 --> 00:00:05.810
is a fairly typical
Sn2 reaction.

00:00:05.810 --> 00:00:07.640
We see this hydroxide.

00:00:07.640 --> 00:00:09.730
It substitutes this bromine.

00:00:09.730 --> 00:00:11.190
It becomes a hydroxyl group.

00:00:11.190 --> 00:00:16.570
The bromine takes-- so it gives
this pair to form this bond.

00:00:16.570 --> 00:00:20.660
Then this bond in both of those
electrons go to the bromine,

00:00:20.660 --> 00:00:23.660
and now that becomes
the bromide anion.

00:00:23.660 --> 00:00:26.220
So we've seen this
multiple times.

00:00:26.220 --> 00:00:28.280
But what I want to really
study in this video,

00:00:28.280 --> 00:00:32.680
is what happens to the
handedness of these molecules

00:00:32.680 --> 00:00:34.700
or to the molecule
right over here,

00:00:34.700 --> 00:00:36.430
I guess you could
say to the substrate,

00:00:36.430 --> 00:00:40.580
although obviously, one group
gets substituted with another.

00:00:40.580 --> 00:00:42.820
So let's name this
thing right over here.

00:00:42.820 --> 00:00:47.980
So if we start counting at
this carbon-- 1, 2, 3, 4.

00:00:47.980 --> 00:00:51.600
Four-carbon backbone,
it is all single bonds.

00:00:51.600 --> 00:00:52.485
This is a butane.

00:00:57.180 --> 00:01:00.750
And then it has a bromine
at the number 2 carbon.

00:01:00.750 --> 00:01:02.618
So this is 2-bromobutane.

00:01:02.618 --> 00:01:04.159
So let me write
this-- 2-bromobutane.

00:01:08.730 --> 00:01:12.230
And then after this
reaction, it becomes--

00:01:12.230 --> 00:01:15.180
so once again, we still
have four carbons,

00:01:15.180 --> 00:01:16.940
but now, instead
of a bromyl group,

00:01:16.940 --> 00:01:19.530
we have a hydroxyl group,
which makes this an alcohol.

00:01:19.530 --> 00:01:26.330
So instead of just
calling it butane,

00:01:26.330 --> 00:01:30.630
we would now call
it butanol because

00:01:30.630 --> 00:01:32.420
of this hydroxyl
group right over here.

00:01:32.420 --> 00:01:34.260
Butanol right here,
and the hydroxyl group

00:01:34.260 --> 00:01:37.080
is attached to the 1, 2.

00:01:37.080 --> 00:01:42.259
So this is 1, 2 carbon,
So this is 2-butanol.

00:01:42.259 --> 00:01:43.800
But what I really
want to think about

00:01:43.800 --> 00:01:45.930
is the chirality,
or the handedness.

00:01:45.930 --> 00:01:48.620
And we see here, this
carbon, in both cases,

00:01:48.620 --> 00:01:50.190
it is a chiral carbon.

00:01:50.190 --> 00:01:52.210
The four things
that it is bonded to

00:01:52.210 --> 00:01:56.570
are all different-- a bromine,
a CH3, a hydrogen, and then

00:01:56.570 --> 00:01:59.280
you could kind of view
this as a ethyl group.

00:01:59.280 --> 00:02:01.630
It's bonded to two
carbons right over here.

00:02:01.630 --> 00:02:02.960
So these are all different.

00:02:02.960 --> 00:02:04.501
So we can think
about its handedness.

00:02:04.501 --> 00:02:07.340
Same here, bonded to
four different things.

00:02:07.340 --> 00:02:09.570
So whenever you're
thinking about chirality,

00:02:09.570 --> 00:02:12.390
the convention is take
the thing that it's

00:02:12.390 --> 00:02:14.490
bonded to that has the
lowest atomic number

00:02:14.490 --> 00:02:15.940
and put that in the back.

00:02:15.940 --> 00:02:18.450
Well, hydrogen has the
lowest atomic number

00:02:18.450 --> 00:02:20.037
out of all of these four things.

00:02:20.037 --> 00:02:22.120
So you could imagine that
being put into the back.

00:02:22.120 --> 00:02:23.828
And in fact, it's
already drawn that way.

00:02:23.828 --> 00:02:26.650
So it's kind of going
into the screen.

00:02:26.650 --> 00:02:30.310
So we'll focus on these
three things right out here.

00:02:30.310 --> 00:02:33.040
And the convention is,
start with the thing that

00:02:33.040 --> 00:02:35.211
has the highest atomic number.

00:02:35.211 --> 00:02:37.210
And the thing that has
the highest atomic number

00:02:37.210 --> 00:02:41.570
is the bromine out
of the four things

00:02:41.570 --> 00:02:43.930
that this chiral
carbon is bonded to,

00:02:43.930 --> 00:02:46.760
and then go in the
order-- and then

00:02:46.760 --> 00:02:49.839
rotate in the order of where the
next highest atomic number is.

00:02:49.839 --> 00:02:52.380
Well, you say, well, you have
a carbon there, a carbon there.

00:02:52.380 --> 00:02:54.080
They have the same
atomic number.

00:02:54.080 --> 00:02:56.960
But then you go to what it's
bonded to for the tie breaker.

00:02:56.960 --> 00:02:58.570
And this is bonded
to three hydrogens.

00:02:58.570 --> 00:03:01.100
This is bonded to two hydrogens
and then another carbon.

00:03:01.100 --> 00:03:04.300
So this extra carbon breaks
that atomic number tiebreaker

00:03:04.300 --> 00:03:06.910
so we go in this direction.

00:03:06.910 --> 00:03:10.090
We go in this direction
right over here.

00:03:10.090 --> 00:03:12.590
And the way I think about it
is what's happening at the top.

00:03:12.590 --> 00:03:14.240
Well, we're moving to the right.

00:03:14.240 --> 00:03:16.570
We are moving in the
clockwise directions.

00:03:16.570 --> 00:03:20.192
So we would call this
R-- R-2-bromobutane.

00:03:20.192 --> 00:03:22.650
And we have other videos, where
we go into much more detail

00:03:22.650 --> 00:03:26.310
about naming chiral molecules.

00:03:26.310 --> 00:03:28.560
But I like to think about
what's happening at the top.

00:03:28.560 --> 00:03:31.340
R, my little mnemonic in
my head is R for right.

00:03:31.340 --> 00:03:34.970
R-2-bromobutane is
right over here.

00:03:34.970 --> 00:03:37.010
Now let's think about
what's going on over here.

00:03:37.010 --> 00:03:39.200
Once again, put the
hydrogen in the back.

00:03:39.200 --> 00:03:41.820
It has the lowest atomic number.

00:03:41.820 --> 00:03:44.630
And so we'll focus on the
carbons and the oxygen.

00:03:44.630 --> 00:03:46.180
Out of carbons
and oxygen, oxygen

00:03:46.180 --> 00:03:48.140
has the highest atomic number.

00:03:50.826 --> 00:03:53.200
And then we'll move in the
direction of the next highest.

00:03:53.200 --> 00:03:56.360
Well, once again, the
CH2CH3, the C's have

00:03:56.360 --> 00:03:58.420
the same atomic number,
but then the tiebreaker

00:03:58.420 --> 00:04:00.080
is when you think about
what it's bonded to.

00:04:00.080 --> 00:04:01.455
This is bonded to
another carbon,

00:04:01.455 --> 00:04:03.070
while this is all hydrogens.

00:04:03.070 --> 00:04:09.714
So we move in this direction,
or counterclockwise direction.

00:04:09.714 --> 00:04:11.130
In our chirality
naming convention

00:04:11.130 --> 00:04:14.070
is we would say
S, S for sinister.

00:04:14.070 --> 00:04:16.820
S-2-butanol.

00:04:16.820 --> 00:04:20.750
Now, the whole reason why I
went through this exercise

00:04:20.750 --> 00:04:26.170
is to show you that it is
Sn2 reaction, the handedness,

00:04:26.170 --> 00:04:29.730
the chirality has
changed for the molecule.

00:04:29.730 --> 00:04:31.930
You can imagine that
this-- if you imagine

00:04:31.930 --> 00:04:34.540
that over here
this 2-bromobutane,

00:04:34.540 --> 00:04:36.881
if this was an umbrella,
where the bromine is kind

00:04:36.881 --> 00:04:39.380
of the handle and this is kind
of an upward-turned umbrella,

00:04:39.380 --> 00:04:42.140
all of a sudden the
nucleophile comes in,

00:04:42.140 --> 00:04:47.200
the leaving group pops off,
and then the umbrella pops off

00:04:47.200 --> 00:04:49.940
or the umbrella goes
in the other direction.

00:04:49.940 --> 00:04:51.350
It has kind of changed.

00:04:51.350 --> 00:04:54.220
Because the nucleophile's
coming from the left side,

00:04:54.220 --> 00:04:56.100
the leaving group is
leaving from the right,

00:04:56.100 --> 00:04:58.647
it has changed its handedness.

00:04:58.647 --> 00:05:00.230
And the reason why
this is interesting

00:05:00.230 --> 00:05:02.180
is, well, OK, it's
an SN2 reaction.

00:05:02.180 --> 00:05:03.100
This now changed.

00:05:03.100 --> 00:05:06.922
I went from R to S.
In an Sn1 reaction,

00:05:06.922 --> 00:05:08.630
as we'll see, because
in an Sn1, reaction

00:05:08.630 --> 00:05:10.540
the nucleophile
leaves on its own,

00:05:10.540 --> 00:05:13.110
And then the nucleo-- sorry,
the leaving group leaves

00:05:13.110 --> 00:05:15.375
on its own, and then the
nucleophile comes in,

00:05:15.375 --> 00:05:17.740
and it could come
in from any side.

00:05:17.740 --> 00:05:20.080
So with an Sn1
reaction, you get a mix.

00:05:20.080 --> 00:05:22.560
You might switch
handedness or you might not

00:05:22.560 --> 00:05:23.330
switch handedness.

00:05:23.330 --> 00:05:24.150
So the whole point
of this video is

00:05:24.150 --> 00:05:26.524
to get you a little bit more--
thinking a little bit more

00:05:26.524 --> 00:05:28.500
about this about the
stereochemistry chemistry

00:05:28.500 --> 00:05:32.078
and the handedness
of Sn2 reactions.