[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.39,0:00:05.68,Default,,0000,0000,0000,,This video is about infinite\Nsequences and their limits.
Dialogue: 0,0:00:06.20,0:00:08.40,Default,,0000,0000,0000,,We'll start by revising what a
Dialogue: 0,0:00:08.40,0:00:13.39,Default,,0000,0000,0000,,simple sequences. You should\Nhave seen that a simple sequence
Dialogue: 0,0:00:13.39,0:00:17.50,Default,,0000,0000,0000,,is a finite list of numbers,\Nlike this one.
Dialogue: 0,0:00:17.50,0:00:22.25,Default,,0000,0000,0000,,It could be something\Nlike 135.
Dialogue: 0,0:00:23.30,0:00:27.12,Default,,0000,0000,0000,,So on up to 19
Dialogue: 0,0:00:27.12,0:00:34.53,Default,,0000,0000,0000,,say. Another possible\Nexample would be something
Dialogue: 0,0:00:34.53,0:00:37.52,Default,,0000,0000,0000,,like four 916.
Dialogue: 0,0:00:38.44,0:00:41.04,Default,,0000,0000,0000,,Perhaps stopping it somewhere
Dialogue: 0,0:00:41.04,0:00:46.06,Default,,0000,0000,0000,,like 81? The numbers in the\Nsequence. I called the terms of
Dialogue: 0,0:00:46.06,0:00:51.09,Default,,0000,0000,0000,,the sequence. So in the second\Nexample here, we would say that
Dialogue: 0,0:00:51.09,0:00:52.92,Default,,0000,0000,0000,,four is the first term.
Dialogue: 0,0:00:53.11,0:00:57.28,Default,,0000,0000,0000,,And nine is the second\Nterm.
Dialogue: 0,0:00:58.86,0:00:59.61,Default,,0000,0000,0000,,And so on.
Dialogue: 0,0:01:01.31,0:01:06.58,Default,,0000,0000,0000,,An infinite sequence like simple\Nsequence is a list of numbers.
Dialogue: 0,0:01:07.29,0:01:09.40,Default,,0000,0000,0000,,But an infinite sequence goes on
Dialogue: 0,0:01:09.40,0:01:16.35,Default,,0000,0000,0000,,forever. So an infinite sequence\Ncould be something like 258.
Dialogue: 0,0:01:17.15,0:01:20.27,Default,,0000,0000,0000,,And this time the terms just\Nkeep on going.
Dialogue: 0,0:01:20.78,0:01:24.32,Default,,0000,0000,0000,,Now, if you see three
Dialogue: 0,0:01:24.32,0:01:28.88,Default,,0000,0000,0000,,dots. Followed by something that\Njust means I've left at some of
Dialogue: 0,0:01:28.88,0:01:30.72,Default,,0000,0000,0000,,the terms, so that will indicate
Dialogue: 0,0:01:30.72,0:01:33.10,Default,,0000,0000,0000,,a finite sequence. But if you
Dialogue: 0,0:01:33.10,0:01:36.27,Default,,0000,0000,0000,,see three dots. And nothing\Nafter them that indicates
Dialogue: 0,0:01:36.27,0:01:39.32,Default,,0000,0000,0000,,the terms go on forever. So\Nthat's an infinite sequence.
Dialogue: 0,0:01:40.43,0:01:45.02,Default,,0000,0000,0000,,We say two sequences at the same\Nif all the terms of the same.
Dialogue: 0,0:01:46.15,0:01:49.87,Default,,0000,0000,0000,,This means that the sequences\Nmust contain the same numbers in
Dialogue: 0,0:01:49.87,0:01:50.88,Default,,0000,0000,0000,,the same places.
Dialogue: 0,0:01:51.75,0:01:59.14,Default,,0000,0000,0000,,So if I have an infinite\Nsequence like 1234 and so on.
Dialogue: 0,0:01:59.76,0:02:07.28,Default,,0000,0000,0000,,This is not the same\Nas the sequence that goes
Dialogue: 0,0:02:07.28,0:02:08.69,Default,,0000,0000,0000,,2143. And so on.
Dialogue: 0,0:02:09.32,0:02:12.54,Default,,0000,0000,0000,,Because even though the sequence\Nhas the same numbers, the
Dialogue: 0,0:02:12.54,0:02:16.08,Default,,0000,0000,0000,,numbers aren't falling in the\Nsame place is so these sequences
Dialogue: 0,0:02:16.08,0:02:17.37,Default,,0000,0000,0000,,are not the same.
Dialogue: 0,0:02:18.54,0:02:21.43,Default,,0000,0000,0000,,The first 2 sequences are\Nwritten here have nice
Dialogue: 0,0:02:21.43,0:02:24.64,Default,,0000,0000,0000,,obvious rules for getting the\NNTH term of the sequence.
Dialogue: 0,0:02:25.98,0:02:31.29,Default,,0000,0000,0000,,So you get the first term in the\Nfirst sequence. You take 1 * 2
Dialogue: 0,0:02:31.29,0:02:33.41,Default,,0000,0000,0000,,and takeaway one that gives you
Dialogue: 0,0:02:33.41,0:02:39.11,Default,,0000,0000,0000,,one. To get the second term you\Ntake 2 * 2 to get the four and
Dialogue: 0,0:02:39.11,0:02:43.16,Default,,0000,0000,0000,,take away one. And this rule\Nwill work for every term of the
Dialogue: 0,0:02:43.16,0:02:44.69,Default,,0000,0000,0000,,sequence. So we say the NTH
Dialogue: 0,0:02:44.69,0:02:48.13,Default,,0000,0000,0000,,term. Is 2 N minus one.
Dialogue: 0,0:02:49.96,0:02:56.14,Default,,0000,0000,0000,,Similarly. The NTH term of\Nthe second sequence here is N
Dialogue: 0,0:02:56.14,0:02:57.77,Default,,0000,0000,0000,,plus one all squared.
Dialogue: 0,0:02:59.20,0:03:03.64,Default,,0000,0000,0000,,The infinite sequence here\Nalso has a rule for getting
Dialogue: 0,0:03:03.64,0:03:04.97,Default,,0000,0000,0000,,the NTH term.
Dialogue: 0,0:03:06.21,0:03:12.50,Default,,0000,0000,0000,,Here we take the number of the\Nterm multiplied by three and
Dialogue: 0,0:03:12.50,0:03:14.07,Default,,0000,0000,0000,,take off 1.
Dialogue: 0,0:03:15.11,0:03:16.18,Default,,0000,0000,0000,,So the NTH term.
Dialogue: 0,0:03:17.24,0:03:21.04,Default,,0000,0000,0000,,Is 3 N minus one.
Dialogue: 0,0:03:22.19,0:03:27.15,Default,,0000,0000,0000,,But not all sequences have a\Nrule for getting the NTH term.
Dialogue: 0,0:03:28.01,0:03:30.93,Default,,0000,0000,0000,,We can have a sequence that\Nlooks really random like.
Dialogue: 0,0:03:31.47,0:03:38.07,Default,,0000,0000,0000,,I'd say root 3\N- 599.7 and so
Dialogue: 0,0:03:38.07,0:03:42.32,Default,,0000,0000,0000,,on. Now, there's certainly no\Nobvious rule for getting the
Dialogue: 0,0:03:42.32,0:03:45.28,Default,,0000,0000,0000,,NTH term for the sequence, but\Nit's still a sequence.
Dialogue: 0,0:03:47.42,0:03:49.76,Default,,0000,0000,0000,,Now let's look at some\Nnotation for sequences.
Dialogue: 0,0:03:50.97,0:03:54.17,Default,,0000,0000,0000,,A common way to the notice\Nsequence is to write the NTH
Dialogue: 0,0:03:54.17,0:03:58.11,Default,,0000,0000,0000,,term in brackets. So for the\Nfinite sequence, the first one
Dialogue: 0,0:03:58.11,0:04:02.19,Default,,0000,0000,0000,,here. The NTH term\Nis 2 and minus one.
Dialogue: 0,0:04:03.26,0:04:05.17,Default,,0000,0000,0000,,So we write that in brackets.
Dialogue: 0,0:04:05.86,0:04:09.60,Default,,0000,0000,0000,,And we also need to show how\Nmany terms the sequence has.
Dialogue: 0,0:04:10.58,0:04:13.98,Default,,0000,0000,0000,,So we say the sequence runs from\NN equals 1.
Dialogue: 0,0:04:15.17,0:04:17.82,Default,,0000,0000,0000,,And the last term in the\Nsequence happens to be the
Dialogue: 0,0:04:17.82,0:04:20.95,Default,,0000,0000,0000,,10th term, so we put a tent\Nup here to show that the
Dialogue: 0,0:04:20.95,0:04:22.40,Default,,0000,0000,0000,,last term is the 10th term.
Dialogue: 0,0:04:23.64,0:04:27.05,Default,,0000,0000,0000,,Here. For the second sequence.
Dialogue: 0,0:04:27.70,0:04:33.58,Default,,0000,0000,0000,,The rule is N plus one squared,\Nso we want that all in brackets.
Dialogue: 0,0:04:34.22,0:04:37.16,Default,,0000,0000,0000,,And this time the sequence runs\Nfrom N equals 1.
Dialogue: 0,0:04:38.36,0:04:42.18,Default,,0000,0000,0000,,Up to that's the eighth term, so\Nwe put Nate here.
Dialogue: 0,0:04:45.22,0:04:47.78,Default,,0000,0000,0000,,We denote the infinite sequence\Nin a similar way.
Dialogue: 0,0:04:48.59,0:04:50.42,Default,,0000,0000,0000,,Again, we put the NTH term in
Dialogue: 0,0:04:50.42,0:04:55.93,Default,,0000,0000,0000,,brackets. So that's three N\Nminus one, all in brackets.
Dialogue: 0,0:04:56.84,0:04:59.95,Default,,0000,0000,0000,,And again we started the first\Ntime, so that's an equals 1.
Dialogue: 0,0:05:01.18,0:05:04.18,Default,,0000,0000,0000,,But to show that the\Nsequence goes on forever, we
Dialogue: 0,0:05:04.18,0:05:05.38,Default,,0000,0000,0000,,put an Infinity here.
Dialogue: 0,0:05:07.00,0:05:13.99,Default,,0000,0000,0000,,From now on will\Njust focus on infinite
Dialogue: 0,0:05:13.99,0:05:18.75,Default,,0000,0000,0000,,sequences. We're often very\Ninterested in what happens to a
Dialogue: 0,0:05:18.75,0:05:21.80,Default,,0000,0000,0000,,sequence as N gets large. There\Nare three particularly
Dialogue: 0,0:05:21.80,0:05:23.50,Default,,0000,0000,0000,,interesting things will look at.
Dialogue: 0,0:05:24.10,0:05:28.06,Default,,0000,0000,0000,,We look at first of
Dialogue: 0,0:05:28.06,0:05:31.92,Default,,0000,0000,0000,,all sequences. That tends
Dialogue: 0,0:05:31.92,0:05:38.84,Default,,0000,0000,0000,,to Infinity.\NAlso
Dialogue: 0,0:05:38.84,0:05:46.24,Default,,0000,0000,0000,,sequences.\NNot
Dialogue: 0,0:05:46.24,0:05:49.61,Default,,0000,0000,0000,,10s. Minus
Dialogue: 0,0:05:49.61,0:05:54.43,Default,,0000,0000,0000,,Infinity.\NAnd
Dialogue: 0,0:05:54.43,0:05:57.46,Default,,0000,0000,0000,,finally,
Dialogue: 0,0:05:57.46,0:06:04.90,Default,,0000,0000,0000,,sequences. That\Ntends to a real limit.
Dialogue: 0,0:06:07.91,0:06:14.97,Default,,0000,0000,0000,,First we look at\Nsequences that tend to
Dialogue: 0,0:06:14.97,0:06:22.63,Default,,0000,0000,0000,,Infinity. We say a\Nsequence tends to Infinity.
Dialogue: 0,0:06:23.16,0:06:26.16,Default,,0000,0000,0000,,If however, large number I\Nchoose, the sequence will
Dialogue: 0,0:06:26.16,0:06:29.82,Default,,0000,0000,0000,,eventually get bigger than that\Nnumber and stay bigger than that
Dialogue: 0,0:06:29.82,0:06:32.13,Default,,0000,0000,0000,,number. So for plot a graph to
Dialogue: 0,0:06:32.13,0:06:35.56,Default,,0000,0000,0000,,show you what I mean. You\Ncan see a sequence tending
Dialogue: 0,0:06:35.56,0:06:36.18,Default,,0000,0000,0000,,to Infinity.
Dialogue: 0,0:06:37.76,0:06:40.06,Default,,0000,0000,0000,,So what I'll do here is.
Dialogue: 0,0:06:41.53,0:06:43.67,Default,,0000,0000,0000,,A put the values of N.
Dialogue: 0,0:06:44.39,0:06:46.34,Default,,0000,0000,0000,,On the X axis.
Dialogue: 0,0:06:46.94,0:06:53.95,Default,,0000,0000,0000,,And then I'll put the value of\Nthe NTH term of the sequence on
Dialogue: 0,0:06:53.95,0:06:55.46,Default,,0000,0000,0000,,the Y axis.
Dialogue: 0,0:06:56.10,0:06:58.76,Default,,0000,0000,0000,,So a sequence that tends\Nto Infinity looks
Dialogue: 0,0:06:58.76,0:06:59.76,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:07:02.09,0:07:09.50,Default,,0000,0000,0000,,Now the terms here are\Ngetting larger and larger and
Dialogue: 0,0:07:09.50,0:07:16.91,Default,,0000,0000,0000,,larger, and I've hit the\Npoint from going off the
Dialogue: 0,0:07:16.91,0:07:19.13,Default,,0000,0000,0000,,page now, but.
Dialogue: 0,0:07:19.91,0:07:24.93,Default,,0000,0000,0000,,If I could draw a line anywhere\Nparallel to the X axis.
Dialogue: 0,0:07:25.79,0:07:26.71,Default,,0000,0000,0000,,Like this one.
Dialogue: 0,0:07:28.60,0:07:33.06,Default,,0000,0000,0000,,Then for the sequence to tend to\NInfinity, we need the terms
Dialogue: 0,0:07:33.06,0:07:37.16,Default,,0000,0000,0000,,eventually to go above that and\Nstay above that. It doesn't
Dialogue: 0,0:07:37.16,0:07:41.25,Default,,0000,0000,0000,,matter how large number I\Nchoose, these terms must go and
Dialogue: 0,0:07:41.25,0:07:45.34,Default,,0000,0000,0000,,stay above that number for the\Nsequence to tend to Infinity.
Dialogue: 0,0:07:45.64,0:07:50.64,Default,,0000,0000,0000,,Here's an example of a sequence\Nthat tends to Infinity.
Dialogue: 0,0:07:51.48,0:07:55.44,Default,,0000,0000,0000,,Will have the sequence\Nthat's an squared going from
Dialogue: 0,0:07:55.44,0:07:57.64,Default,,0000,0000,0000,,N equals 1 to Infinity.
Dialogue: 0,0:07:58.81,0:08:05.07,Default,,0000,0000,0000,,So that starts off going\N1, four 916.
Dialogue: 0,0:08:05.20,0:08:07.77,Default,,0000,0000,0000,,And so on.
Dialogue: 0,0:08:07.77,0:08:11.30,Default,,0000,0000,0000,,So I can plot a graph of the
Dialogue: 0,0:08:11.30,0:08:16.99,Default,,0000,0000,0000,,sequence. I won't bother putting\Nin the valleys friend. I'll just
Dialogue: 0,0:08:16.99,0:08:20.20,Default,,0000,0000,0000,,plots the values of the terms of
Dialogue: 0,0:08:20.20,0:08:22.38,Default,,0000,0000,0000,,the sequence. So.
Dialogue: 0,0:08:23.05,0:08:28.05,Default,,0000,0000,0000,,The sequence. Will look\Nsomething like this.
Dialogue: 0,0:08:29.33,0:08:33.63,Default,,0000,0000,0000,,Now you can see that however\Nlarge number I choose, the terms
Dialogue: 0,0:08:33.63,0:08:37.56,Default,,0000,0000,0000,,of the sequence will definitely\Ngo above and stay above it,
Dialogue: 0,0:08:37.56,0:08:41.14,Default,,0000,0000,0000,,because this sequence keeps on\Nincreasing and it increases very
Dialogue: 0,0:08:41.14,0:08:44.01,Default,,0000,0000,0000,,fast. So this sequence\Ndefinitely tends to Infinity.
Dialogue: 0,0:08:45.56,0:08:51.41,Default,,0000,0000,0000,,Now, even if a sequence\Nsometimes goes down, it can
Dialogue: 0,0:08:51.41,0:08:53.75,Default,,0000,0000,0000,,still tend to Infinity.
Dialogue: 0,0:08:54.99,0:08:57.49,Default,,0000,0000,0000,,Sequence that looks a bit\Nlike this.
Dialogue: 0,0:08:58.92,0:09:04.04,Default,,0000,0000,0000,,Starts of small goes up for\Nawhile, comes back down and then
Dialogue: 0,0:09:04.04,0:09:06.18,Default,,0000,0000,0000,,goes up for awhile again.
Dialogue: 0,0:09:06.68,0:09:08.68,Default,,0000,0000,0000,,Comes down not so far.
Dialogue: 0,0:09:09.19,0:09:10.49,Default,,0000,0000,0000,,And carries on going up.
Dialogue: 0,0:09:11.79,0:09:14.91,Default,,0000,0000,0000,,Now this sequence does sometimes
Dialogue: 0,0:09:14.91,0:09:20.00,Default,,0000,0000,0000,,come down. But it always goes\Nup again and it would always
Dialogue: 0,0:09:20.00,0:09:23.89,Default,,0000,0000,0000,,get above any number I choose\Nand it will always stay above
Dialogue: 0,0:09:23.89,0:09:27.45,Default,,0000,0000,0000,,that as well. So this sequence\Nalso tends to Infinity, even
Dialogue: 0,0:09:27.45,0:09:28.75,Default,,0000,0000,0000,,though it decreases sometimes.
Dialogue: 0,0:09:30.73,0:09:36.41,Default,,0000,0000,0000,,Now here's a sequence that\Ndoesn't tend to Infinity, even
Dialogue: 0,0:09:36.41,0:09:39.25,Default,,0000,0000,0000,,though it always gets bigger.
Dialogue: 0,0:09:40.42,0:09:47.21,Default,,0000,0000,0000,,Will start off with\Nthe first time, the
Dialogue: 0,0:09:47.21,0:09:49.76,Default,,0000,0000,0000,,sequence being 0.
Dialogue: 0,0:09:50.64,0:09:54.13,Default,,0000,0000,0000,,And then a lot of 100. So this\Nis a very big scale here.
Dialogue: 0,0:09:54.85,0:10:00.37,Default,,0000,0000,0000,,But here. Then I'll add on half\Nof the hundreds or out on 50.
Dialogue: 0,0:10:01.35,0:10:04.46,Default,,0000,0000,0000,,Aladdin half of 50 which is 25.
Dialogue: 0,0:10:05.15,0:10:07.88,Default,,0000,0000,0000,,It'll keep adding on half the\Nprevious Mount I did.
Dialogue: 0,0:10:08.49,0:10:10.41,Default,,0000,0000,0000,,And this sequence.
Dialogue: 0,0:10:10.93,0:10:12.99,Default,,0000,0000,0000,,Does this kind of thing?
Dialogue: 0,0:10:13.51,0:10:19.11,Default,,0000,0000,0000,,And The thing is, it never ever\Ngets above 200.
Dialogue: 0,0:10:21.01,0:10:24.59,Default,,0000,0000,0000,,So we found a number here that\Nthe sequence doesn't go above
Dialogue: 0,0:10:24.59,0:10:27.86,Default,,0000,0000,0000,,and stay above, so that must\Nmean the sequence doesn't tend
Dialogue: 0,0:10:27.86,0:10:34.99,Default,,0000,0000,0000,,to Infinity. And finally, here's\Nan example of a sequence that
Dialogue: 0,0:10:34.99,0:10:38.18,Default,,0000,0000,0000,,doesn't tend to Infinity, even
Dialogue: 0,0:10:38.18,0:10:41.47,Default,,0000,0000,0000,,though. It gets really,\Nreally big.
Dialogue: 0,0:10:42.71,0:10:47.79,Default,,0000,0000,0000,,Will start off with the first\Ntime being 0 again, then one,
Dialogue: 0,0:10:47.79,0:10:50.32,Default,,0000,0000,0000,,then zero, then two, then zero,
Dialogue: 0,0:10:50.32,0:10:53.60,Default,,0000,0000,0000,,then 3. And so on.
Dialogue: 0,0:10:54.73,0:11:00.47,Default,,0000,0000,0000,,Now eventually the sequence will\Nget above any number I choose.
Dialogue: 0,0:11:01.27,0:11:05.82,Default,,0000,0000,0000,,But it never stays above because\Nit always goes back to zero, and
Dialogue: 0,0:11:05.82,0:11:09.32,Default,,0000,0000,0000,,because it doesn't stay above\Nany number, I choose, the
Dialogue: 0,0:11:09.32,0:11:10.72,Default,,0000,0000,0000,,sequence doesn't tend to
Dialogue: 0,0:11:10.72,0:11:16.69,Default,,0000,0000,0000,,Infinity. Now we look at\Nsequences that tends to
Dialogue: 0,0:11:16.69,0:11:17.87,Default,,0000,0000,0000,,minus Infinity.
Dialogue: 0,0:11:19.16,0:11:22.52,Default,,0000,0000,0000,,We say a sequence tends to\Nminus Infinity If however
Dialogue: 0,0:11:22.52,0:11:25.88,Default,,0000,0000,0000,,large as negative a number,\NI choose the sequence goes
Dialogue: 0,0:11:25.88,0:11:27.90,Default,,0000,0000,0000,,below it and stays below it.
Dialogue: 0,0:11:28.95,0:11:33.73,Default,,0000,0000,0000,,So a good example is something\Nlike the sequence minus N cubed.
Dialogue: 0,0:11:34.31,0:11:36.56,Default,,0000,0000,0000,,From N equals 1 to Infinity.
Dialogue: 0,0:11:37.19,0:11:39.15,Default,,0000,0000,0000,,I can sketch a graph of this.
Dialogue: 0,0:11:39.71,0:11:45.96,Default,,0000,0000,0000,,This starts off\Nat minus one.
Dialogue: 0,0:11:46.49,0:11:49.90,Default,,0000,0000,0000,,And then falls really rapidly.
Dialogue: 0,0:11:51.46,0:11:54.79,Default,,0000,0000,0000,,So however low an\Numbrella choose.
Dialogue: 0,0:11:56.70,0:11:58.67,Default,,0000,0000,0000,,The sequence goes below it and
Dialogue: 0,0:11:58.67,0:12:02.71,Default,,0000,0000,0000,,stays below it. So this sequence\Ntends to minus Infinity.
Dialogue: 0,0:12:03.71,0:12:08.73,Default,,0000,0000,0000,,Just cause sequence goes\Nbelow any number doesn't mean
Dialogue: 0,0:12:08.73,0:12:15.43,Default,,0000,0000,0000,,it tends to minus Infinity.\NIt has to stay below it. So
Dialogue: 0,0:12:15.43,0:12:17.66,Default,,0000,0000,0000,,a sequence like this.
Dialogue: 0,0:12:20.29,0:12:27.14,Default,,0000,0000,0000,,Which goes minus 1 + 1 -\N2 + 2 and so on.
Dialogue: 0,0:12:29.69,0:12:33.78,Default,,0000,0000,0000,,Now, even though the terms of\Nthe sequence go below any large
Dialogue: 0,0:12:33.78,0:12:37.19,Default,,0000,0000,0000,,negative number I choose, they\Ndon't stay below because we
Dialogue: 0,0:12:37.19,0:12:38.90,Default,,0000,0000,0000,,always get a positive term
Dialogue: 0,0:12:38.90,0:12:43.00,Default,,0000,0000,0000,,again. So this sequence doesn't\Ntend to minus Infinity.
Dialogue: 0,0:12:43.60,0:12:46.57,Default,,0000,0000,0000,,In fact, the sequence doesn't\Ntend to any limit at all.
Dialogue: 0,0:12:47.68,0:12:55.01,Default,,0000,0000,0000,,If the sequence tends to\Nminus Infinity, we write it
Dialogue: 0,0:12:55.01,0:13:01.51,Default,,0000,0000,0000,,like this. We\Nwrite XN Arrow Minus
Dialogue: 0,0:13:01.51,0:13:08.75,Default,,0000,0000,0000,,Infinity. As end tends\Nto Infinity or the limit of
Dialogue: 0,0:13:08.75,0:13:12.47,Default,,0000,0000,0000,,XN. As I intend to
Dialogue: 0,0:13:12.47,0:13:15.76,Default,,0000,0000,0000,,Infinity. Equals minus Infinity.
Dialogue: 0,0:13:17.48,0:13:24.81,Default,,0000,0000,0000,,Finally, we'll look at\Nsequences that tend to
Dialogue: 0,0:13:24.81,0:13:27.56,Default,,0000,0000,0000,,a real limit.
Dialogue: 0,0:13:29.12,0:13:33.67,Default,,0000,0000,0000,,We say a sequence tends to a\Nreal limit if there's a number,
Dialogue: 0,0:13:33.67,0:13:35.07,Default,,0000,0000,0000,,which I'll call L.
Dialogue: 0,0:13:36.00,0:13:36.93,Default,,0000,0000,0000,,So that's.
Dialogue: 0,0:13:39.34,0:13:43.77,Default,,0000,0000,0000,,The sequence gets closer and\Ncloser to L and stays very
Dialogue: 0,0:13:43.77,0:13:48.21,Default,,0000,0000,0000,,close to it, so a sequence\Ntending to L might look
Dialogue: 0,0:13:48.21,0:13:49.42,Default,,0000,0000,0000,,something like this.
Dialogue: 0,0:13:56.41,0:14:01.02,Default,,0000,0000,0000,,Now what I mean by getting\Ncloser and staying close to L is
Dialogue: 0,0:14:01.02,0:14:04.93,Default,,0000,0000,0000,,however small an interval I\Nchoose around all. So let's say
Dialogue: 0,0:14:04.93,0:14:06.70,Default,,0000,0000,0000,,I pick this tiny interval.
Dialogue: 0,0:14:13.25,0:14:16.45,Default,,0000,0000,0000,,The sequence must eventually get\Ninside that interval and stay
Dialogue: 0,0:14:16.45,0:14:19.65,Default,,0000,0000,0000,,inside the interval. It doesn't\Neven matter if the sequence
Dialogue: 0,0:14:19.65,0:14:24.13,Default,,0000,0000,0000,,doesn't actually ever hit al, so\Nlong as it gets as close as we
Dialogue: 0,0:14:24.13,0:14:27.97,Default,,0000,0000,0000,,like to. Ellen stays as close as\Nwe like, then that sequence
Dialogue: 0,0:14:27.97,0:14:35.38,Default,,0000,0000,0000,,tends to L. Here's an\Nexample of a sequence that
Dialogue: 0,0:14:35.38,0:14:38.53,Default,,0000,0000,0000,,has a real limit.
Dialogue: 0,0:14:39.41,0:14:43.86,Default,,0000,0000,0000,,Will have the sequence\Nbeing one over N for N
Dialogue: 0,0:14:43.86,0:14:45.64,Default,,0000,0000,0000,,equals 1 to Infinity.
Dialogue: 0,0:14:46.73,0:14:49.70,Default,,0000,0000,0000,,I'll sketch a graph of this.
Dialogue: 0,0:14:50.23,0:14:53.65,Default,,0000,0000,0000,,The first time the sequence
Dialogue: 0,0:14:53.65,0:15:00.18,Default,,0000,0000,0000,,is one. Then it goes 1/2,\Nthen it goes to 3rd in the
Dialogue: 0,0:15:00.18,0:15:01.84,Default,,0000,0000,0000,,quarter and so on.
Dialogue: 0,0:15:02.37,0:15:08.08,Default,,0000,0000,0000,,I can see this sequence gets\Ncloser and closer to 0.
Dialogue: 0,0:15:08.62,0:15:12.17,Default,,0000,0000,0000,,And if I pick any tiny\Nnumber, the sequence will
Dialogue: 0,0:15:12.17,0:15:16.08,Default,,0000,0000,0000,,eventually get that close to\N0 and it will stay that
Dialogue: 0,0:15:16.08,0:15:19.98,Default,,0000,0000,0000,,close 'cause it keeps going\Ndown. So this sequence has a
Dialogue: 0,0:15:19.98,0:15:22.82,Default,,0000,0000,0000,,real limit and that real\Nlimit is 0.
Dialogue: 0,0:15:24.58,0:15:30.81,Default,,0000,0000,0000,,Allow autograph of a\Nsequence that tends to
Dialogue: 0,0:15:30.81,0:15:33.15,Default,,0000,0000,0000,,real limit 3.
Dialogue: 0,0:15:34.26,0:15:38.99,Default,,0000,0000,0000,,So\Nput
Dialogue: 0,0:15:38.99,0:15:43.73,Default,,0000,0000,0000,,three\Nhere.
Dialogue: 0,0:15:44.88,0:15:48.30,Default,,0000,0000,0000,,And I'll show you the intervals\NI can choose.
Dialogue: 0,0:16:00.65,0:16:03.32,Default,,0000,0000,0000,,Now I can choose a sequence.
Dialogue: 0,0:16:04.05,0:16:06.65,Default,,0000,0000,0000,,That sometimes goes away from 3.
Dialogue: 0,0:16:07.96,0:16:11.48,Default,,0000,0000,0000,,But eventually see it
Dialogue: 0,0:16:11.48,0:16:15.24,Default,,0000,0000,0000,,gets trapped. And this
Dialogue: 0,0:16:15.24,0:16:19.83,Default,,0000,0000,0000,,larger interval. Then it gets\Ntrapped in the smaller interval.
Dialogue: 0,0:16:22.20,0:16:23.67,Default,,0000,0000,0000,,And whatever interval I drew.
Dialogue: 0,0:16:24.58,0:16:26.07,Default,,0000,0000,0000,,It would eventually get trapped.
Dialogue: 0,0:16:26.66,0:16:29.44,Default,,0000,0000,0000,,That close to 3.
Dialogue: 0,0:16:30.01,0:16:32.72,Default,,0000,0000,0000,,So even though this sequence\Nseems to go all over the place.
Dialogue: 0,0:16:33.33,0:16:37.00,Default,,0000,0000,0000,,It eventually gets as close\Nas we like to three and stays
Dialogue: 0,0:16:37.00,0:16:39.45,Default,,0000,0000,0000,,that close, so this sequence\Ntends to three.
Dialogue: 0,0:16:41.14,0:16:48.71,Default,,0000,0000,0000,,But if a sequence tends to\Nreal limit L, we write it
Dialogue: 0,0:16:48.71,0:16:54.20,Default,,0000,0000,0000,,like this. We say XN\Ntends to L.
Dialogue: 0,0:16:55.06,0:17:02.56,Default,,0000,0000,0000,,As an tends to Infinity or\Nthe limit of XN equals L.
Dialogue: 0,0:17:03.51,0:17:05.15,Default,,0000,0000,0000,,A Zen tends to Infinity.
Dialogue: 0,0:17:06.92,0:17:14.38,Default,,0000,0000,0000,,If a sequence doesn't tend\Nto a real limit, we
Dialogue: 0,0:17:14.38,0:17:16.62,Default,,0000,0000,0000,,say it's divergent.
Dialogue: 0,0:17:18.02,0:17:22.03,Default,,0000,0000,0000,,So sequences that tend to\NInfinity and minus Infinity
Dialogue: 0,0:17:22.03,0:17:23.37,Default,,0000,0000,0000,,are all divergent.
Dialogue: 0,0:17:24.52,0:17:27.50,Default,,0000,0000,0000,,But there are some sequences\Nthat don't tend to either plus
Dialogue: 0,0:17:27.50,0:17:29.13,Default,,0000,0000,0000,,or minus Infinity that are still
Dialogue: 0,0:17:29.13,0:17:31.05,Default,,0000,0000,0000,,divergent. Here's an example.
Dialogue: 0,0:17:32.50,0:17:39.71,Default,,0000,0000,0000,,Will have the sequence going\N012, one 0 - 1
Dialogue: 0,0:17:39.71,0:17:43.32,Default,,0000,0000,0000,,- 2 - 1 zero
Dialogue: 0,0:17:43.32,0:17:47.50,Default,,0000,0000,0000,,and then. Repeating\Nitself like that.
Dialogue: 0,0:17:49.09,0:17:50.71,Default,,0000,0000,0000,,And so on.
Dialogue: 0,0:17:51.49,0:17:54.11,Default,,0000,0000,0000,,Now, this sequence certainly\Ndoesn't get closer and stay
Dialogue: 0,0:17:54.11,0:17:57.60,Default,,0000,0000,0000,,closer to any real number, so it\Ndoesn't have a real limit.
Dialogue: 0,0:17:58.24,0:18:00.24,Default,,0000,0000,0000,,But it doesn't go off to plus or
Dialogue: 0,0:18:00.24,0:18:03.73,Default,,0000,0000,0000,,minus Infinity either. So\Nthis sequence is divergent,
Dialogue: 0,0:18:03.73,0:18:06.58,Default,,0000,0000,0000,,but doesn't tend to plus or\Nminus Infinity.
Dialogue: 0,0:18:07.68,0:18:11.26,Default,,0000,0000,0000,,Now this sequence keeps on\Nrepeating itself will repeat
Dialogue: 0,0:18:11.26,0:18:15.26,Default,,0000,0000,0000,,itself forever. The sequence\Nlike this is called periodic.
Dialogue: 0,0:18:15.79,0:18:19.44,Default,,0000,0000,0000,,And periodic sequences are\Na good example of divergent
Dialogue: 0,0:18:19.44,0:18:19.84,Default,,0000,0000,0000,,sequences.