0:00:00.551,0:00:02.050
So what we're going[br]to attempt to do

0:00:02.050,0:00:05.570
is evaluate this[br]sum right over here,

0:00:05.570,0:00:08.109
evaluate what this[br]series is, negative 2

0:00:08.109,0:00:11.660
over n plus 1 times n plus[br]2, starting at n equals 2,

0:00:11.660,0:00:13.750
all the way to infinity.

0:00:13.750,0:00:16.850
And if we wanted to see what[br]this looks like, it starts at n

0:00:16.850,0:00:17.400
equals 2.

0:00:17.400,0:00:20.430
So when n equals 2, this[br]is negative 2 over 2

0:00:20.430,0:00:24.370
plus 1, which is 3, times[br]2 plus 2, which is 4.

0:00:24.370,0:00:28.940
Then when n is equal to 3,[br]this is negative 2 over 3

0:00:28.940,0:00:32.560
plus 1, which is 4, times[br]3 plus 2, which is 5.

0:00:32.560,0:00:34.820
And it just keeps going[br]like that, negative 2

0:00:34.820,0:00:37.670
over 5 times 6.

0:00:37.670,0:00:40.620
And it just keeps[br]going on and on and on.

0:00:40.620,0:00:43.880
And now, it looks pretty clear[br]that each successive term

0:00:43.880,0:00:45.620
is getting smaller.

0:00:45.620,0:00:47.680
And it's getting[br]smaller reasonably fast.

0:00:47.680,0:00:51.814
So it's reasonable to assume[br]that even though you have

0:00:51.814,0:00:53.480
an infinite number[br]of terms, it actually

0:00:53.480,0:00:54.842
might give you a finite value.

0:00:54.842,0:00:56.800
But it doesn't jump out[br]at me, at least the way

0:00:56.800,0:00:58.470
that I've looked[br]at it right now,

0:00:58.470,0:00:59.970
as to what this sum[br]would be, or how

0:00:59.970,0:01:02.120
to actually figure out that sum.

0:01:02.120,0:01:04.269
So what I want you to do[br]now is pause this video.

0:01:04.269,0:01:07.630
And I'm going to give you a hint[br]about how to think about this.

0:01:07.630,0:01:12.170
Try to dig up your memories[br]of partial fraction expansion,

0:01:12.170,0:01:14.160
or partial fraction[br]decomposition,

0:01:14.160,0:01:17.920
to turn this expression into[br]the sum of two fractions.

0:01:17.920,0:01:22.280
And that might help us think[br]about what this sum is.

0:01:22.280,0:01:24.337
So I'm assuming you've[br]given a go at it.

0:01:24.337,0:01:25.920
So let's try to[br]manipulate this thing.

0:01:25.920,0:01:28.840
And let's see if we can rewrite[br]this as a sum of two fractions.

0:01:28.840,0:01:32.870
So this is negative[br]2 over-- and I'm

0:01:32.870,0:01:35.990
going to do this in two[br]different colors-- n plus 1

0:01:35.990,0:01:37.650
times n plus 2.

0:01:40.160,0:01:42.870
And we remember from our[br]partial fraction expansion

0:01:42.870,0:01:45.710
that we can rewrite this as[br]the sum of two fractions,

0:01:45.710,0:01:54.846
as A over n plus 1[br]plus B over n plus 2.

0:01:54.846,0:01:55.970
And why is this reasonable?

0:01:55.970,0:01:57.090
Well, if you're[br]adding two fractions,

0:01:57.090,0:01:58.620
you want to find a[br]common denominator, which

0:01:58.620,0:02:00.650
would be a multiple of[br]the two denominators.

0:02:00.650,0:02:03.300
This is clearly a multiple of[br]both of these denominators.

0:02:03.300,0:02:05.660
And we learned in partial[br]fraction expansion

0:02:05.660,0:02:08.919
that whatever we have up here,[br]especially because the degree

0:02:08.919,0:02:13.020
here is lower than the degree[br]down here, whatever we have

0:02:13.020,0:02:15.890
up here is going to be a degree[br]lower than what we have here.

0:02:15.890,0:02:18.000
So this is a first-degree[br]term in terms of n,

0:02:18.000,0:02:20.850
so these are going to be[br]constant terms up here.

0:02:20.850,0:02:22.740
So let's figure out[br]what A and B are.

0:02:22.740,0:02:25.580
So if we perform[br]the addition-- well,

0:02:25.580,0:02:27.590
let's just rewrite[br]both of these with

0:02:27.590,0:02:29.400
the same common denominator.

0:02:29.400,0:02:34.190
So let's rewrite[br]A over n plus 1,

0:02:34.190,0:02:37.720
but let's multiply the numerator[br]and denominator by n plus 2.

0:02:37.720,0:02:41.570
So we multiply the numerator[br]by n plus 2 and the denominator

0:02:41.570,0:02:42.100
by n plus 2.

0:02:42.100,0:02:44.490
I haven't changed the value[br]of this first fraction.

0:02:44.490,0:02:50.850
Similarly, let's do the same[br]thing with B over n plus 2.

0:02:50.850,0:02:54.470
Multiply the numerator and the[br]denominator by n plus 1, so

0:02:54.470,0:02:57.960
n plus 1 over n plus 1.

0:02:57.960,0:03:01.276
Once again, I haven't change[br]the value of this fraction.

0:03:01.276,0:03:03.400
But by doing this, I now[br]have a common denominator,

0:03:03.400,0:03:04.570
and I can add.

0:03:04.570,0:03:12.714
So this is going to be equal[br]to n plus 1 times n plus 2 is

0:03:12.714,0:03:13.380
our denominator.

0:03:15.920,0:03:19.690
And then our numerator--[br]let me expand it out.

0:03:19.690,0:03:21.690
This is going to be,[br]if I distribute the A,

0:03:21.690,0:03:25.290
it is An plus 2A.

0:03:25.290,0:03:31.730
So let me write[br]that, An plus 2A.

0:03:31.730,0:03:40.680
And then let's distribute[br]this B, plus Bn plus B.

0:03:40.680,0:03:42.680
Now, what I want to do[br]is I want to rewrite this

0:03:42.680,0:03:44.330
so I have all of the n terms.

0:03:44.330,0:03:51.070
So for example, An plus Bn--[br]I could factor an n out.

0:03:51.070,0:03:58.730
And I could rewrite that[br]as A plus B times n, those

0:03:58.730,0:04:00.350
two terms right over there.

0:04:00.350,0:04:03.895
And then these two[br]terms, the 2A plus B,

0:04:03.895,0:04:09.080
I could just write it[br]like this, plus 2A plus B.

0:04:09.080,0:04:17.550
And, of course, all of that is[br]over n plus 1 times n plus 2.

0:04:20.915,0:04:24.020
So how do we solve for A and B?

0:04:24.020,0:04:26.650
Well, the realization[br]is that this thing

0:04:26.650,0:04:29.070
must be equal to negative 2.

0:04:29.070,0:04:31.879
These two things must[br]be equal to each other.

0:04:31.879,0:04:33.670
Remember, we're making[br]the claim that this,

0:04:33.670,0:04:36.160
which is the same thing[br]as this, is equal to this.

0:04:36.160,0:04:38.754
That's the whole reason[br]why we started doing this.

0:04:38.754,0:04:40.420
So we're making the[br]claim that these two

0:04:40.420,0:04:42.680
things are equivalent.

0:04:42.680,0:04:44.470
We're making this claim.

0:04:44.470,0:04:47.560
So everything in the numerator[br]must be equal to negative 2.

0:04:47.560,0:04:48.710
So how do we do that?

0:04:48.710,0:04:52.130
It looks like we have[br]two unknowns here.

0:04:52.130,0:04:54.930
To figure out two unknowns, we[br]normally need two equations.

0:04:54.930,0:04:56.990
Well, the realization[br]here is, look,

0:04:56.990,0:05:00.030
we have an n term on[br]the left-hand side here.

0:05:00.030,0:05:01.520
We have no n term here.

0:05:01.520,0:05:03.950
So you literally could view[br]this, instead of just viewing

0:05:03.950,0:05:05.366
this as negative[br]2, you could view

0:05:05.366,0:05:10.960
this as negative 2 plus[br]0n, plus 0 times n.

0:05:10.960,0:05:11.740
That's not "on."

0:05:11.740,0:05:17.599
That's 0-- let me write[br]it this way-- 0 times n.

0:05:17.599,0:05:19.140
So when you look at[br]it this way, it's

0:05:19.140,0:05:22.430
clear that A plus B is[br]the coefficient on n.

0:05:22.430,0:05:24.700
That must be equal to 0.

0:05:24.700,0:05:28.340
A plus B must be equal to 0.

0:05:28.340,0:05:30.810
And this is kind of[br]bread-and-butter partial

0:05:30.810,0:05:32.180
fraction expansion.

0:05:32.180,0:05:34.530
We have other videos on that[br]if you need to review that.

0:05:34.530,0:05:43.010
And the constant part, 2A plus[br]B, is equal to negative 2.

0:05:46.100,0:05:51.020
And so now we have two[br]equations in two unknowns.

0:05:51.020,0:05:53.020
And we could solve it a[br]bunch of different ways.

0:05:53.020,0:05:55.450
But one interesting way is[br]let's multiply the top equation

0:05:55.450,0:05:56.930
by negative 1.

0:05:56.930,0:06:01.090
So then this becomes negative[br]A minus B is equal to-- well,

0:06:01.090,0:06:03.360
negative 1 times 0 is still 0.

0:06:03.360,0:06:05.600
Now we can add these[br]two things together.

0:06:05.600,0:06:11.110
And we are left with 2A minus[br]A is A, plus B minus B-- well,

0:06:11.110,0:06:13.830
those cancel out.

0:06:13.830,0:06:16.320
A is equal to negative 2.

0:06:16.320,0:06:20.120
And if A is equal to[br]negative 2, A plus B is 0,

0:06:20.120,0:06:21.520
B must be equal to 2.

0:06:24.090,0:06:28.000
Negative 2 plus 2 is equal to 0.

0:06:28.000,0:06:31.100
We solved for A. And then I[br]substituted it back up here.

0:06:31.100,0:06:34.860
So now we can rewrite all[br]of this right over here.

0:06:34.860,0:06:37.830
We can rewrite it as[br]the sum-- and actually,

0:06:37.830,0:06:39.200
let me do a little bit instead.

0:06:39.200,0:06:43.020
Let me just write it as[br]a finite sum as opposed

0:06:43.020,0:06:44.200
to an infinite sum.

0:06:44.200,0:06:47.450
And then we can just take the[br]limit as we go to infinity.

0:06:47.450,0:06:49.190
So let me rewrite it like this.

0:06:49.190,0:06:53.700
So this is the sum from n[br]equals 2-- instead to infinity,

0:06:53.700,0:06:56.540
I'll just say to capital[br]N. And then later, we

0:06:56.540,0:07:00.750
could take the limit as this[br]goes to infinity of-- well,

0:07:00.750,0:07:03.850
instead of writing this, I can[br]write this right over here.

0:07:03.850,0:07:06.370
So A is negative 2.

0:07:06.370,0:07:11.110
So it's negative[br]2 over n plus 1.

0:07:11.110,0:07:17.820
And then B is 2,[br]plus B over n plus 2.

0:07:17.820,0:07:20.610
So once again, I've just[br]expressed this as a finite sum.

0:07:20.610,0:07:23.210
Later, we can take the limit[br]as capital N approaches

0:07:23.210,0:07:25.020
infinity to figure out[br]what this thing is.

0:07:25.020,0:07:27.870
Oh, sorry, and B-- let[br]me not write B anymore.

0:07:27.870,0:07:33.450
We now know that B[br]is 2 over n plus 2.

0:07:33.450,0:07:37.850
Now, how does this actually[br]go about helping us?

0:07:37.850,0:07:39.330
Well, let's do what[br]we did up here.

0:07:39.330,0:07:42.260
Let's actually write out what[br]this is going to be equal to.

0:07:42.260,0:07:46.900
This is going to be[br]equal to-- when n is 2,

0:07:46.900,0:07:54.410
this is negative 2/3, so[br]it's negative 2/3, plus 2/4.

0:08:00.160,0:08:02.810
So that's n equals-- let me[br]do it down here, because I'm

0:08:02.810,0:08:04.400
about to run out of real estate.

0:08:04.400,0:08:06.640
That is when n is equal to 2.

0:08:06.640,0:08:10.120
Now, what about when[br]n is equal to 3?

0:08:10.120,0:08:21.920
When n is equal to 3, this[br]is going to be negative 2/4

0:08:21.920,0:08:22.610
plus 2/5.

0:08:28.770,0:08:30.530
What about when n is equal to 4?

0:08:30.530,0:08:33.890
I think you might see a pattern[br]that's starting to form.

0:08:33.890,0:08:34.640
Let's do one more.

0:08:34.640,0:08:42.090
When n is equal to[br]4, well, then, this

0:08:42.090,0:08:46.710
is going to be[br]negative 2/5-- let

0:08:46.710,0:08:53.170
me do that same blue color--[br]negative 2/5 plus 2/6.

0:08:57.640,0:09:00.460
And we're just[br]going to keep going.

0:09:00.460,0:09:02.520
Let me scroll down to[br]get some space-- we're

0:09:02.520,0:09:05.190
going to keep going all the[br]way until the N-th term.

0:09:08.960,0:09:13.680
So plus dot dot dot plus[br]our capital N-th term,

0:09:13.680,0:09:24.110
which is going to be negative[br]2 over capital N plus 1 plus 2

0:09:24.110,0:09:27.560
over capital N plus 2.

0:09:27.560,0:09:29.310
So I think you might[br]see the pattern here.

0:09:29.310,0:09:33.466
Notice, from our first when[br]n equals 2, we got the 2/4.

0:09:33.466,0:09:35.590
But then when n equals 3,[br]you had the negative 2/4.

0:09:35.590,0:09:37.450
That cancels with that.

0:09:37.450,0:09:39.090
When n equals 3, you had 2/5.

0:09:39.090,0:09:42.690
Then that cancels when n[br]equals 4 with the negative 2/5.

0:09:42.690,0:09:47.170
So the second term cancels[br]with-- the second part,

0:09:47.170,0:09:50.380
I guess, for each[br]n, for each index,

0:09:50.380,0:09:53.040
cancels out with the first[br]part for the next index.

0:09:53.040,0:09:55.420
And so that's just going to[br]keep happening all the way

0:09:55.420,0:09:59.730
until n is equal to capital N.

0:09:59.730,0:10:02.300
And so this is going to[br]cancel out with the one right

0:10:02.300,0:10:03.110
before it.

0:10:03.110,0:10:06.540
And all we're going[br]to be left with

0:10:06.540,0:10:13.790
is this term and this[br]term right over here.

0:10:13.790,0:10:15.950
So let's rewrite that.

0:10:15.950,0:10:19.090
So we get-- let's[br]get more space here.

0:10:19.090,0:10:26.290
This thing can be rewritten[br]as the sum from lowercase n

0:10:26.290,0:10:30.850
equals 2 to capital[br]N of negative 2

0:10:30.850,0:10:37.020
over n plus 1 plus[br]2 over n plus 2

0:10:37.020,0:10:39.440
is equal to-- well, everything[br]else in the middle canceled

0:10:39.440,0:10:39.940
out.

0:10:39.940,0:10:43.740
We're just left[br]with negative 2/3

0:10:43.740,0:10:50.460
plus 2 over capital N plus 2.

0:10:50.460,0:10:53.110
So this was a huge[br]simplification right over here.

0:10:53.110,0:10:57.000
And remember, our original sum[br]that we wanted to calculate,

0:10:57.000,0:11:00.510
that just has a limit as[br]capital N goes to infinity.

0:11:00.510,0:11:04.620
So let's just take the limit[br]as capital N goes to infinity.

0:11:04.620,0:11:06.170
So let me write it this way.

0:11:06.170,0:11:07.878
Well, actually, let[br]me write it this way.

0:11:07.878,0:11:10.810
The limit-- so we can[br]write it this way.

0:11:10.810,0:11:15.350
The limit as[br]capital N approaches

0:11:15.350,0:11:20.030
infinity is going to be equal[br]to the limit as capital N

0:11:20.030,0:11:22.239
approaches infinity[br]of-- well, we just

0:11:22.239,0:11:23.280
figured out what this is.

0:11:23.280,0:11:33.240
This is negative 2/3 plus[br]2 over capital N plus 2.

0:11:33.240,0:11:36.180
Well, as n goes to[br]infinity, this negative 2/3

0:11:36.180,0:11:37.600
doesn't get impacted at all.

0:11:37.600,0:11:40.375
This term right over here, 2[br]over an ever larger number,

0:11:40.375,0:11:42.000
over an infinitely[br]large number-- well,

0:11:42.000,0:11:43.520
that's going to go to 0.

0:11:43.520,0:11:47.990
And we're going to be[br]left with negative 2/3.

0:11:47.990,0:11:48.750
And we're done.

0:11:48.750,0:11:54.740
We were able to figure out the[br]sum of this infinite series.

0:11:54.740,0:11:58.460
So this thing right over here[br]is equal to negative 2/3.

0:11:58.460,0:12:01.320
And this type of series is[br]called a telescoping series--

0:12:01.320,0:12:02.920
telescoping, I should say.

0:12:02.920,0:12:04.380
This is a telescoping series.

0:12:09.270,0:12:12.130
And a telescoping series[br]is a general term.

0:12:12.130,0:12:14.500
So if you were to[br]take its partial sums,

0:12:14.500,0:12:18.430
it has this pattern right over[br]here, where, in each term,

0:12:18.430,0:12:20.020
you're starting to[br]cancel things out.

0:12:20.020,0:12:23.270
So what you're left with[br]is just a fixed number

0:12:23.270,0:12:25.520
of terms at the end.

0:12:25.520,0:12:27.130
But either way,[br]this was a pretty--

0:12:27.130,0:12:29.421
it's a little bit hairy, but[br]it was a pretty satisfying

0:12:29.421,0:12:30.640
problem.