[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.55,0:00:02.05,Default,,0000,0000,0000,,So what we're going\Nto attempt to do
Dialogue: 0,0:00:02.05,0:00:05.57,Default,,0000,0000,0000,,is evaluate this\Nsum right over here,
Dialogue: 0,0:00:05.57,0:00:08.11,Default,,0000,0000,0000,,evaluate what this\Nseries is, negative 2
Dialogue: 0,0:00:08.11,0:00:11.66,Default,,0000,0000,0000,,over n plus 1 times n plus\N2, starting at n equals 2,
Dialogue: 0,0:00:11.66,0:00:13.75,Default,,0000,0000,0000,,all the way to infinity.
Dialogue: 0,0:00:13.75,0:00:16.85,Default,,0000,0000,0000,,And if we wanted to see what\Nthis looks like, it starts at n
Dialogue: 0,0:00:16.85,0:00:17.40,Default,,0000,0000,0000,,equals 2.
Dialogue: 0,0:00:17.40,0:00:20.43,Default,,0000,0000,0000,,So when n equals 2, this\Nis negative 2 over 2
Dialogue: 0,0:00:20.43,0:00:24.37,Default,,0000,0000,0000,,plus 1, which is 3, times\N2 plus 2, which is 4.
Dialogue: 0,0:00:24.37,0:00:28.94,Default,,0000,0000,0000,,Then when n is equal to 3,\Nthis is negative 2 over 3
Dialogue: 0,0:00:28.94,0:00:32.56,Default,,0000,0000,0000,,plus 1, which is 4, times\N3 plus 2, which is 5.
Dialogue: 0,0:00:32.56,0:00:34.82,Default,,0000,0000,0000,,And it just keeps going\Nlike that, negative 2
Dialogue: 0,0:00:34.82,0:00:37.67,Default,,0000,0000,0000,,over 5 times 6.
Dialogue: 0,0:00:37.67,0:00:40.62,Default,,0000,0000,0000,,And it just keeps\Ngoing on and on and on.
Dialogue: 0,0:00:40.62,0:00:43.88,Default,,0000,0000,0000,,And now, it looks pretty clear\Nthat each successive term
Dialogue: 0,0:00:43.88,0:00:45.62,Default,,0000,0000,0000,,is getting smaller.
Dialogue: 0,0:00:45.62,0:00:47.68,Default,,0000,0000,0000,,And it's getting\Nsmaller reasonably fast.
Dialogue: 0,0:00:47.68,0:00:51.81,Default,,0000,0000,0000,,So it's reasonable to assume\Nthat even though you have
Dialogue: 0,0:00:51.81,0:00:53.48,Default,,0000,0000,0000,,an infinite number\Nof terms, it actually
Dialogue: 0,0:00:53.48,0:00:54.84,Default,,0000,0000,0000,,might give you a finite value.
Dialogue: 0,0:00:54.84,0:00:56.80,Default,,0000,0000,0000,,But it doesn't jump out\Nat me, at least the way
Dialogue: 0,0:00:56.80,0:00:58.47,Default,,0000,0000,0000,,that I've looked\Nat it right now,
Dialogue: 0,0:00:58.47,0:00:59.97,Default,,0000,0000,0000,,as to what this sum\Nwould be, or how
Dialogue: 0,0:00:59.97,0:01:02.12,Default,,0000,0000,0000,,to actually figure out that sum.
Dialogue: 0,0:01:02.12,0:01:04.27,Default,,0000,0000,0000,,So what I want you to do\Nnow is pause this video.
Dialogue: 0,0:01:04.27,0:01:07.63,Default,,0000,0000,0000,,And I'm going to give you a hint\Nabout how to think about this.
Dialogue: 0,0:01:07.63,0:01:12.17,Default,,0000,0000,0000,,Try to dig up your memories\Nof partial fraction expansion,
Dialogue: 0,0:01:12.17,0:01:14.16,Default,,0000,0000,0000,,or partial fraction\Ndecomposition,
Dialogue: 0,0:01:14.16,0:01:17.92,Default,,0000,0000,0000,,to turn this expression into\Nthe sum of two fractions.
Dialogue: 0,0:01:17.92,0:01:22.28,Default,,0000,0000,0000,,And that might help us think\Nabout what this sum is.
Dialogue: 0,0:01:22.28,0:01:24.34,Default,,0000,0000,0000,,So I'm assuming you've\Ngiven a go at it.
Dialogue: 0,0:01:24.34,0:01:25.92,Default,,0000,0000,0000,,So let's try to\Nmanipulate this thing.
Dialogue: 0,0:01:25.92,0:01:28.84,Default,,0000,0000,0000,,And let's see if we can rewrite\Nthis as a sum of two fractions.
Dialogue: 0,0:01:28.84,0:01:32.87,Default,,0000,0000,0000,,So this is negative\N2 over-- and I'm
Dialogue: 0,0:01:32.87,0:01:35.99,Default,,0000,0000,0000,,going to do this in two\Ndifferent colors-- n plus 1
Dialogue: 0,0:01:35.99,0:01:37.65,Default,,0000,0000,0000,,times n plus 2.
Dialogue: 0,0:01:40.16,0:01:42.87,Default,,0000,0000,0000,,And we remember from our\Npartial fraction expansion
Dialogue: 0,0:01:42.87,0:01:45.71,Default,,0000,0000,0000,,that we can rewrite this as\Nthe sum of two fractions,
Dialogue: 0,0:01:45.71,0:01:54.85,Default,,0000,0000,0000,,as A over n plus 1\Nplus B over n plus 2.
Dialogue: 0,0:01:54.85,0:01:55.97,Default,,0000,0000,0000,,And why is this reasonable?
Dialogue: 0,0:01:55.97,0:01:57.09,Default,,0000,0000,0000,,Well, if you're\Nadding two fractions,
Dialogue: 0,0:01:57.09,0:01:58.62,Default,,0000,0000,0000,,you want to find a\Ncommon denominator, which
Dialogue: 0,0:01:58.62,0:02:00.65,Default,,0000,0000,0000,,would be a multiple of\Nthe two denominators.
Dialogue: 0,0:02:00.65,0:02:03.30,Default,,0000,0000,0000,,This is clearly a multiple of\Nboth of these denominators.
Dialogue: 0,0:02:03.30,0:02:05.66,Default,,0000,0000,0000,,And we learned in partial\Nfraction expansion
Dialogue: 0,0:02:05.66,0:02:08.92,Default,,0000,0000,0000,,that whatever we have up here,\Nespecially because the degree
Dialogue: 0,0:02:08.92,0:02:13.02,Default,,0000,0000,0000,,here is lower than the degree\Ndown here, whatever we have
Dialogue: 0,0:02:13.02,0:02:15.89,Default,,0000,0000,0000,,up here is going to be a degree\Nlower than what we have here.
Dialogue: 0,0:02:15.89,0:02:18.00,Default,,0000,0000,0000,,So this is a first-degree\Nterm in terms of n,
Dialogue: 0,0:02:18.00,0:02:20.85,Default,,0000,0000,0000,,so these are going to be\Nconstant terms up here.
Dialogue: 0,0:02:20.85,0:02:22.74,Default,,0000,0000,0000,,So let's figure out\Nwhat A and B are.
Dialogue: 0,0:02:22.74,0:02:25.58,Default,,0000,0000,0000,,So if we perform\Nthe addition-- well,
Dialogue: 0,0:02:25.58,0:02:27.59,Default,,0000,0000,0000,,let's just rewrite\Nboth of these with
Dialogue: 0,0:02:27.59,0:02:29.40,Default,,0000,0000,0000,,the same common denominator.
Dialogue: 0,0:02:29.40,0:02:34.19,Default,,0000,0000,0000,,So let's rewrite\NA over n plus 1,
Dialogue: 0,0:02:34.19,0:02:37.72,Default,,0000,0000,0000,,but let's multiply the numerator\Nand denominator by n plus 2.
Dialogue: 0,0:02:37.72,0:02:41.57,Default,,0000,0000,0000,,So we multiply the numerator\Nby n plus 2 and the denominator
Dialogue: 0,0:02:41.57,0:02:42.10,Default,,0000,0000,0000,,by n plus 2.
Dialogue: 0,0:02:42.10,0:02:44.49,Default,,0000,0000,0000,,I haven't changed the value\Nof this first fraction.
Dialogue: 0,0:02:44.49,0:02:50.85,Default,,0000,0000,0000,,Similarly, let's do the same\Nthing with B over n plus 2.
Dialogue: 0,0:02:50.85,0:02:54.47,Default,,0000,0000,0000,,Multiply the numerator and the\Ndenominator by n plus 1, so
Dialogue: 0,0:02:54.47,0:02:57.96,Default,,0000,0000,0000,,n plus 1 over n plus 1.
Dialogue: 0,0:02:57.96,0:03:01.28,Default,,0000,0000,0000,,Once again, I haven't change\Nthe value of this fraction.
Dialogue: 0,0:03:01.28,0:03:03.40,Default,,0000,0000,0000,,But by doing this, I now\Nhave a common denominator,
Dialogue: 0,0:03:03.40,0:03:04.57,Default,,0000,0000,0000,,and I can add.
Dialogue: 0,0:03:04.57,0:03:12.71,Default,,0000,0000,0000,,So this is going to be equal\Nto n plus 1 times n plus 2 is
Dialogue: 0,0:03:12.71,0:03:13.38,Default,,0000,0000,0000,,our denominator.
Dialogue: 0,0:03:15.92,0:03:19.69,Default,,0000,0000,0000,,And then our numerator--\Nlet me expand it out.
Dialogue: 0,0:03:19.69,0:03:21.69,Default,,0000,0000,0000,,This is going to be,\Nif I distribute the A,
Dialogue: 0,0:03:21.69,0:03:25.29,Default,,0000,0000,0000,,it is An plus 2A.
Dialogue: 0,0:03:25.29,0:03:31.73,Default,,0000,0000,0000,,So let me write\Nthat, An plus 2A.
Dialogue: 0,0:03:31.73,0:03:40.68,Default,,0000,0000,0000,,And then let's distribute\Nthis B, plus Bn plus B.
Dialogue: 0,0:03:40.68,0:03:42.68,Default,,0000,0000,0000,,Now, what I want to do\Nis I want to rewrite this
Dialogue: 0,0:03:42.68,0:03:44.33,Default,,0000,0000,0000,,so I have all of the n terms.
Dialogue: 0,0:03:44.33,0:03:51.07,Default,,0000,0000,0000,,So for example, An plus Bn--\NI could factor an n out.
Dialogue: 0,0:03:51.07,0:03:58.73,Default,,0000,0000,0000,,And I could rewrite that\Nas A plus B times n, those
Dialogue: 0,0:03:58.73,0:04:00.35,Default,,0000,0000,0000,,two terms right over there.
Dialogue: 0,0:04:00.35,0:04:03.90,Default,,0000,0000,0000,,And then these two\Nterms, the 2A plus B,
Dialogue: 0,0:04:03.90,0:04:09.08,Default,,0000,0000,0000,,I could just write it\Nlike this, plus 2A plus B.
Dialogue: 0,0:04:09.08,0:04:17.55,Default,,0000,0000,0000,,And, of course, all of that is\Nover n plus 1 times n plus 2.
Dialogue: 0,0:04:20.92,0:04:24.02,Default,,0000,0000,0000,,So how do we solve for A and B?
Dialogue: 0,0:04:24.02,0:04:26.65,Default,,0000,0000,0000,,Well, the realization\Nis that this thing
Dialogue: 0,0:04:26.65,0:04:29.07,Default,,0000,0000,0000,,must be equal to negative 2.
Dialogue: 0,0:04:29.07,0:04:31.88,Default,,0000,0000,0000,,These two things must\Nbe equal to each other.
Dialogue: 0,0:04:31.88,0:04:33.67,Default,,0000,0000,0000,,Remember, we're making\Nthe claim that this,
Dialogue: 0,0:04:33.67,0:04:36.16,Default,,0000,0000,0000,,which is the same thing\Nas this, is equal to this.
Dialogue: 0,0:04:36.16,0:04:38.75,Default,,0000,0000,0000,,That's the whole reason\Nwhy we started doing this.
Dialogue: 0,0:04:38.75,0:04:40.42,Default,,0000,0000,0000,,So we're making the\Nclaim that these two
Dialogue: 0,0:04:40.42,0:04:42.68,Default,,0000,0000,0000,,things are equivalent.
Dialogue: 0,0:04:42.68,0:04:44.47,Default,,0000,0000,0000,,We're making this claim.
Dialogue: 0,0:04:44.47,0:04:47.56,Default,,0000,0000,0000,,So everything in the numerator\Nmust be equal to negative 2.
Dialogue: 0,0:04:47.56,0:04:48.71,Default,,0000,0000,0000,,So how do we do that?
Dialogue: 0,0:04:48.71,0:04:52.13,Default,,0000,0000,0000,,It looks like we have\Ntwo unknowns here.
Dialogue: 0,0:04:52.13,0:04:54.93,Default,,0000,0000,0000,,To figure out two unknowns, we\Nnormally need two equations.
Dialogue: 0,0:04:54.93,0:04:56.99,Default,,0000,0000,0000,,Well, the realization\Nhere is, look,
Dialogue: 0,0:04:56.99,0:05:00.03,Default,,0000,0000,0000,,we have an n term on\Nthe left-hand side here.
Dialogue: 0,0:05:00.03,0:05:01.52,Default,,0000,0000,0000,,We have no n term here.
Dialogue: 0,0:05:01.52,0:05:03.95,Default,,0000,0000,0000,,So you literally could view\Nthis, instead of just viewing
Dialogue: 0,0:05:03.95,0:05:05.37,Default,,0000,0000,0000,,this as negative\N2, you could view
Dialogue: 0,0:05:05.37,0:05:10.96,Default,,0000,0000,0000,,this as negative 2 plus\N0n, plus 0 times n.
Dialogue: 0,0:05:10.96,0:05:11.74,Default,,0000,0000,0000,,That's not "on."
Dialogue: 0,0:05:11.74,0:05:17.60,Default,,0000,0000,0000,,That's 0-- let me write\Nit this way-- 0 times n.
Dialogue: 0,0:05:17.60,0:05:19.14,Default,,0000,0000,0000,,So when you look at\Nit this way, it's
Dialogue: 0,0:05:19.14,0:05:22.43,Default,,0000,0000,0000,,clear that A plus B is\Nthe coefficient on n.
Dialogue: 0,0:05:22.43,0:05:24.70,Default,,0000,0000,0000,,That must be equal to 0.
Dialogue: 0,0:05:24.70,0:05:28.34,Default,,0000,0000,0000,,A plus B must be equal to 0.
Dialogue: 0,0:05:28.34,0:05:30.81,Default,,0000,0000,0000,,And this is kind of\Nbread-and-butter partial
Dialogue: 0,0:05:30.81,0:05:32.18,Default,,0000,0000,0000,,fraction expansion.
Dialogue: 0,0:05:32.18,0:05:34.53,Default,,0000,0000,0000,,We have other videos on that\Nif you need to review that.
Dialogue: 0,0:05:34.53,0:05:43.01,Default,,0000,0000,0000,,And the constant part, 2A plus\NB, is equal to negative 2.
Dialogue: 0,0:05:46.10,0:05:51.02,Default,,0000,0000,0000,,And so now we have two\Nequations in two unknowns.
Dialogue: 0,0:05:51.02,0:05:53.02,Default,,0000,0000,0000,,And we could solve it a\Nbunch of different ways.
Dialogue: 0,0:05:53.02,0:05:55.45,Default,,0000,0000,0000,,But one interesting way is\Nlet's multiply the top equation
Dialogue: 0,0:05:55.45,0:05:56.93,Default,,0000,0000,0000,,by negative 1.
Dialogue: 0,0:05:56.93,0:06:01.09,Default,,0000,0000,0000,,So then this becomes negative\NA minus B is equal to-- well,
Dialogue: 0,0:06:01.09,0:06:03.36,Default,,0000,0000,0000,,negative 1 times 0 is still 0.
Dialogue: 0,0:06:03.36,0:06:05.60,Default,,0000,0000,0000,,Now we can add these\Ntwo things together.
Dialogue: 0,0:06:05.60,0:06:11.11,Default,,0000,0000,0000,,And we are left with 2A minus\NA is A, plus B minus B-- well,
Dialogue: 0,0:06:11.11,0:06:13.83,Default,,0000,0000,0000,,those cancel out.
Dialogue: 0,0:06:13.83,0:06:16.32,Default,,0000,0000,0000,,A is equal to negative 2.
Dialogue: 0,0:06:16.32,0:06:20.12,Default,,0000,0000,0000,,And if A is equal to\Nnegative 2, A plus B is 0,
Dialogue: 0,0:06:20.12,0:06:21.52,Default,,0000,0000,0000,,B must be equal to 2.
Dialogue: 0,0:06:24.09,0:06:28.00,Default,,0000,0000,0000,,Negative 2 plus 2 is equal to 0.
Dialogue: 0,0:06:28.00,0:06:31.10,Default,,0000,0000,0000,,We solved for A. And then I\Nsubstituted it back up here.
Dialogue: 0,0:06:31.10,0:06:34.86,Default,,0000,0000,0000,,So now we can rewrite all\Nof this right over here.
Dialogue: 0,0:06:34.86,0:06:37.83,Default,,0000,0000,0000,,We can rewrite it as\Nthe sum-- and actually,
Dialogue: 0,0:06:37.83,0:06:39.20,Default,,0000,0000,0000,,let me do a little bit instead.
Dialogue: 0,0:06:39.20,0:06:43.02,Default,,0000,0000,0000,,Let me just write it as\Na finite sum as opposed
Dialogue: 0,0:06:43.02,0:06:44.20,Default,,0000,0000,0000,,to an infinite sum.
Dialogue: 0,0:06:44.20,0:06:47.45,Default,,0000,0000,0000,,And then we can just take the\Nlimit as we go to infinity.
Dialogue: 0,0:06:47.45,0:06:49.19,Default,,0000,0000,0000,,So let me rewrite it like this.
Dialogue: 0,0:06:49.19,0:06:53.70,Default,,0000,0000,0000,,So this is the sum from n\Nequals 2-- instead to infinity,
Dialogue: 0,0:06:53.70,0:06:56.54,Default,,0000,0000,0000,,I'll just say to capital\NN. And then later, we
Dialogue: 0,0:06:56.54,0:07:00.75,Default,,0000,0000,0000,,could take the limit as this\Ngoes to infinity of-- well,
Dialogue: 0,0:07:00.75,0:07:03.85,Default,,0000,0000,0000,,instead of writing this, I can\Nwrite this right over here.
Dialogue: 0,0:07:03.85,0:07:06.37,Default,,0000,0000,0000,,So A is negative 2.
Dialogue: 0,0:07:06.37,0:07:11.11,Default,,0000,0000,0000,,So it's negative\N2 over n plus 1.
Dialogue: 0,0:07:11.11,0:07:17.82,Default,,0000,0000,0000,,And then B is 2,\Nplus B over n plus 2.
Dialogue: 0,0:07:17.82,0:07:20.61,Default,,0000,0000,0000,,So once again, I've just\Nexpressed this as a finite sum.
Dialogue: 0,0:07:20.61,0:07:23.21,Default,,0000,0000,0000,,Later, we can take the limit\Nas capital N approaches
Dialogue: 0,0:07:23.21,0:07:25.02,Default,,0000,0000,0000,,infinity to figure out\Nwhat this thing is.
Dialogue: 0,0:07:25.02,0:07:27.87,Default,,0000,0000,0000,,Oh, sorry, and B-- let\Nme not write B anymore.
Dialogue: 0,0:07:27.87,0:07:33.45,Default,,0000,0000,0000,,We now know that B\Nis 2 over n plus 2.
Dialogue: 0,0:07:33.45,0:07:37.85,Default,,0000,0000,0000,,Now, how does this actually\Ngo about helping us?
Dialogue: 0,0:07:37.85,0:07:39.33,Default,,0000,0000,0000,,Well, let's do what\Nwe did up here.
Dialogue: 0,0:07:39.33,0:07:42.26,Default,,0000,0000,0000,,Let's actually write out what\Nthis is going to be equal to.
Dialogue: 0,0:07:42.26,0:07:46.90,Default,,0000,0000,0000,,This is going to be\Nequal to-- when n is 2,
Dialogue: 0,0:07:46.90,0:07:54.41,Default,,0000,0000,0000,,this is negative 2/3, so\Nit's negative 2/3, plus 2/4.
Dialogue: 0,0:08:00.16,0:08:02.81,Default,,0000,0000,0000,,So that's n equals-- let me\Ndo it down here, because I'm
Dialogue: 0,0:08:02.81,0:08:04.40,Default,,0000,0000,0000,,about to run out of real estate.
Dialogue: 0,0:08:04.40,0:08:06.64,Default,,0000,0000,0000,,That is when n is equal to 2.
Dialogue: 0,0:08:06.64,0:08:10.12,Default,,0000,0000,0000,,Now, what about when\Nn is equal to 3?
Dialogue: 0,0:08:10.12,0:08:21.92,Default,,0000,0000,0000,,When n is equal to 3, this\Nis going to be negative 2/4
Dialogue: 0,0:08:21.92,0:08:22.61,Default,,0000,0000,0000,,plus 2/5.
Dialogue: 0,0:08:28.77,0:08:30.53,Default,,0000,0000,0000,,What about when n is equal to 4?
Dialogue: 0,0:08:30.53,0:08:33.89,Default,,0000,0000,0000,,I think you might see a pattern\Nthat's starting to form.
Dialogue: 0,0:08:33.89,0:08:34.64,Default,,0000,0000,0000,,Let's do one more.
Dialogue: 0,0:08:34.64,0:08:42.09,Default,,0000,0000,0000,,When n is equal to\N4, well, then, this
Dialogue: 0,0:08:42.09,0:08:46.71,Default,,0000,0000,0000,,is going to be\Nnegative 2/5-- let
Dialogue: 0,0:08:46.71,0:08:53.17,Default,,0000,0000,0000,,me do that same blue color--\Nnegative 2/5 plus 2/6.
Dialogue: 0,0:08:57.64,0:09:00.46,Default,,0000,0000,0000,,And we're just\Ngoing to keep going.
Dialogue: 0,0:09:00.46,0:09:02.52,Default,,0000,0000,0000,,Let me scroll down to\Nget some space-- we're
Dialogue: 0,0:09:02.52,0:09:05.19,Default,,0000,0000,0000,,going to keep going all the\Nway until the N-th term.
Dialogue: 0,0:09:08.96,0:09:13.68,Default,,0000,0000,0000,,So plus dot dot dot plus\Nour capital N-th term,
Dialogue: 0,0:09:13.68,0:09:24.11,Default,,0000,0000,0000,,which is going to be negative\N2 over capital N plus 1 plus 2
Dialogue: 0,0:09:24.11,0:09:27.56,Default,,0000,0000,0000,,over capital N plus 2.
Dialogue: 0,0:09:27.56,0:09:29.31,Default,,0000,0000,0000,,So I think you might\Nsee the pattern here.
Dialogue: 0,0:09:29.31,0:09:33.47,Default,,0000,0000,0000,,Notice, from our first when\Nn equals 2, we got the 2/4.
Dialogue: 0,0:09:33.47,0:09:35.59,Default,,0000,0000,0000,,But then when n equals 3,\Nyou had the negative 2/4.
Dialogue: 0,0:09:35.59,0:09:37.45,Default,,0000,0000,0000,,That cancels with that.
Dialogue: 0,0:09:37.45,0:09:39.09,Default,,0000,0000,0000,,When n equals 3, you had 2/5.
Dialogue: 0,0:09:39.09,0:09:42.69,Default,,0000,0000,0000,,Then that cancels when n\Nequals 4 with the negative 2/5.
Dialogue: 0,0:09:42.69,0:09:47.17,Default,,0000,0000,0000,,So the second term cancels\Nwith-- the second part,
Dialogue: 0,0:09:47.17,0:09:50.38,Default,,0000,0000,0000,,I guess, for each\Nn, for each index,
Dialogue: 0,0:09:50.38,0:09:53.04,Default,,0000,0000,0000,,cancels out with the first\Npart for the next index.
Dialogue: 0,0:09:53.04,0:09:55.42,Default,,0000,0000,0000,,And so that's just going to\Nkeep happening all the way
Dialogue: 0,0:09:55.42,0:09:59.73,Default,,0000,0000,0000,,until n is equal to capital N.
Dialogue: 0,0:09:59.73,0:10:02.30,Default,,0000,0000,0000,,And so this is going to\Ncancel out with the one right
Dialogue: 0,0:10:02.30,0:10:03.11,Default,,0000,0000,0000,,before it.
Dialogue: 0,0:10:03.11,0:10:06.54,Default,,0000,0000,0000,,And all we're going\Nto be left with
Dialogue: 0,0:10:06.54,0:10:13.79,Default,,0000,0000,0000,,is this term and this\Nterm right over here.
Dialogue: 0,0:10:13.79,0:10:15.95,Default,,0000,0000,0000,,So let's rewrite that.
Dialogue: 0,0:10:15.95,0:10:19.09,Default,,0000,0000,0000,,So we get-- let's\Nget more space here.
Dialogue: 0,0:10:19.09,0:10:26.29,Default,,0000,0000,0000,,This thing can be rewritten\Nas the sum from lowercase n
Dialogue: 0,0:10:26.29,0:10:30.85,Default,,0000,0000,0000,,equals 2 to capital\NN of negative 2
Dialogue: 0,0:10:30.85,0:10:37.02,Default,,0000,0000,0000,,over n plus 1 plus\N2 over n plus 2
Dialogue: 0,0:10:37.02,0:10:39.44,Default,,0000,0000,0000,,is equal to-- well, everything\Nelse in the middle canceled
Dialogue: 0,0:10:39.44,0:10:39.94,Default,,0000,0000,0000,,out.
Dialogue: 0,0:10:39.94,0:10:43.74,Default,,0000,0000,0000,,We're just left\Nwith negative 2/3
Dialogue: 0,0:10:43.74,0:10:50.46,Default,,0000,0000,0000,,plus 2 over capital N plus 2.
Dialogue: 0,0:10:50.46,0:10:53.11,Default,,0000,0000,0000,,So this was a huge\Nsimplification right over here.
Dialogue: 0,0:10:53.11,0:10:57.00,Default,,0000,0000,0000,,And remember, our original sum\Nthat we wanted to calculate,
Dialogue: 0,0:10:57.00,0:11:00.51,Default,,0000,0000,0000,,that just has a limit as\Ncapital N goes to infinity.
Dialogue: 0,0:11:00.51,0:11:04.62,Default,,0000,0000,0000,,So let's just take the limit\Nas capital N goes to infinity.
Dialogue: 0,0:11:04.62,0:11:06.17,Default,,0000,0000,0000,,So let me write it this way.
Dialogue: 0,0:11:06.17,0:11:07.88,Default,,0000,0000,0000,,Well, actually, let\Nme write it this way.
Dialogue: 0,0:11:07.88,0:11:10.81,Default,,0000,0000,0000,,The limit-- so we can\Nwrite it this way.
Dialogue: 0,0:11:10.81,0:11:15.35,Default,,0000,0000,0000,,The limit as\Ncapital N approaches
Dialogue: 0,0:11:15.35,0:11:20.03,Default,,0000,0000,0000,,infinity is going to be equal\Nto the limit as capital N
Dialogue: 0,0:11:20.03,0:11:22.24,Default,,0000,0000,0000,,approaches infinity\Nof-- well, we just
Dialogue: 0,0:11:22.24,0:11:23.28,Default,,0000,0000,0000,,figured out what this is.
Dialogue: 0,0:11:23.28,0:11:33.24,Default,,0000,0000,0000,,This is negative 2/3 plus\N2 over capital N plus 2.
Dialogue: 0,0:11:33.24,0:11:36.18,Default,,0000,0000,0000,,Well, as n goes to\Ninfinity, this negative 2/3
Dialogue: 0,0:11:36.18,0:11:37.60,Default,,0000,0000,0000,,doesn't get impacted at all.
Dialogue: 0,0:11:37.60,0:11:40.38,Default,,0000,0000,0000,,This term right over here, 2\Nover an ever larger number,
Dialogue: 0,0:11:40.38,0:11:42.00,Default,,0000,0000,0000,,over an infinitely\Nlarge number-- well,
Dialogue: 0,0:11:42.00,0:11:43.52,Default,,0000,0000,0000,,that's going to go to 0.
Dialogue: 0,0:11:43.52,0:11:47.99,Default,,0000,0000,0000,,And we're going to be\Nleft with negative 2/3.
Dialogue: 0,0:11:47.99,0:11:48.75,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:11:48.75,0:11:54.74,Default,,0000,0000,0000,,We were able to figure out the\Nsum of this infinite series.
Dialogue: 0,0:11:54.74,0:11:58.46,Default,,0000,0000,0000,,So this thing right over here\Nis equal to negative 2/3.
Dialogue: 0,0:11:58.46,0:12:01.32,Default,,0000,0000,0000,,And this type of series is\Ncalled a telescoping series--
Dialogue: 0,0:12:01.32,0:12:02.92,Default,,0000,0000,0000,,telescoping, I should say.
Dialogue: 0,0:12:02.92,0:12:04.38,Default,,0000,0000,0000,,This is a telescoping series.
Dialogue: 0,0:12:09.27,0:12:12.13,Default,,0000,0000,0000,,And a telescoping series\Nis a general term.
Dialogue: 0,0:12:12.13,0:12:14.50,Default,,0000,0000,0000,,So if you were to\Ntake its partial sums,
Dialogue: 0,0:12:14.50,0:12:18.43,Default,,0000,0000,0000,,it has this pattern right over\Nhere, where, in each term,
Dialogue: 0,0:12:18.43,0:12:20.02,Default,,0000,0000,0000,,you're starting to\Ncancel things out.
Dialogue: 0,0:12:20.02,0:12:23.27,Default,,0000,0000,0000,,So what you're left with\Nis just a fixed number
Dialogue: 0,0:12:23.27,0:12:25.52,Default,,0000,0000,0000,,of terms at the end.
Dialogue: 0,0:12:25.52,0:12:27.13,Default,,0000,0000,0000,,But either way,\Nthis was a pretty--
Dialogue: 0,0:12:27.13,0:12:29.42,Default,,0000,0000,0000,,it's a little bit hairy, but\Nit was a pretty satisfying
Dialogue: 0,0:12:29.42,0:12:30.64,Default,,0000,0000,0000,,problem.