[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.02,0:00:03.72,Default,,0000,0000,0000,,We're going to try to determine\Nwhether this alternating series
Dialogue: 0,0:00:03.72,0:00:08.19,Default,,0000,0000,0000,,converges or diverges\Nusing the alternating series test.
Dialogue: 0,0:00:08.19,0:00:12.11,Default,,0000,0000,0000,,The first thing we have to figure out \Nis actually a formula for b-n.
Dialogue: 0,0:00:12.11,0:00:18.07,Default,,0000,0000,0000,,If you recall, b-n is basically the \Nabsolute value of each of the terms.
Dialogue: 0,0:00:18.07,0:00:19.46,Default,,0000,0000,0000,,And if we're looking at this,
Dialogue: 0,0:00:19.46,0:00:22.94,Default,,0000,0000,0000,,we just ignore the part \Nthat causes the sign to alternate,
Dialogue: 0,0:00:22.94,0:00:28.22,Default,,0000,0000,0000,,Hopefully you would agree\Nthat b-n is just 1 over n.
Dialogue: 0,0:00:28.22,0:00:29.57,Default,,0000,0000,0000,,The first term is 1,
Dialogue: 0,0:00:29.57,0:00:32.49,Default,,0000,0000,0000,,the second term we're \Nsubtracting is one half [1/2],
Dialogue: 0,0:00:32.49,0:00:33.60,Default,,0000,0000,0000,,and then one third [1/3],
Dialogue: 0,0:00:33.60,0:00:37.14,Default,,0000,0000,0000,,then we subtract one fourth [1/4], \Nand we add one fifth [1/5].
Dialogue: 0,0:00:37.14,0:00:40.35,Default,,0000,0000,0000,,That would be our expression for b-n.
Dialogue: 0,0:00:40.35,0:00:45.87,Default,,0000,0000,0000,,There are two conditions \Nthat need to be met for this test.
Dialogue: 0,0:00:45.87,0:00:51.39,Default,,0000,0000,0000,,So the first condition is \Nthat the n plus first term
Dialogue: 0,0:00:51.39,0:00:55.37,Default,,0000,0000,0000,,is supposed to be less than \Nor equal to the nth term
Dialogue: 0,0:00:55.37,0:00:58.86,Default,,0000,0000,0000,,for all values of n beyond a certain point.
Dialogue: 0,0:00:58.86,0:01:04.51,Default,,0000,0000,0000,,And to test that, we just have to figure \Nout what the n plus first term would be;
Dialogue: 0,0:01:04.51,0:01:08.30,Default,,0000,0000,0000,,of course, that would be 1 over n plus 1.
Dialogue: 0,0:01:08.30,0:01:11.91,Default,,0000,0000,0000,,If we compare that to the nth term, \Nwhich is 1 over n,
Dialogue: 0,0:01:11.91,0:01:17.44,Default,,0000,0000,0000,,clearly, 1 over n plus 1 \Nis less than 1 over n,
Dialogue: 0,0:01:17.44,0:01:20.49,Default,,0000,0000,0000,,so that satisfies the first condition.
Dialogue: 0,0:01:20.49,0:01:30.56,Default,,0000,0000,0000,,The second condition is that the limit as \Nn goes to infinity for b-n needs to equal 0,
Dialogue: 0,0:01:30.56,0:01:32.42,Default,,0000,0000,0000,,so that's the next thing to test.
Dialogue: 0,0:01:32.42,0:01:36.42,Default,,0000,0000,0000,,And in some examples, \Nwe'll actually do this first
Dialogue: 0,0:01:36.42,0:01:40.66,Default,,0000,0000,0000,,because if this is not true, \Nthen the whole test is going to fail.
Dialogue: 0,0:01:40.66,0:01:47.07,Default,,0000,0000,0000,,But in this case, if we look at the limit \Nas n goes to infinity for 1 over n,
Dialogue: 0,0:01:47.07,0:01:51.54,Default,,0000,0000,0000,,hopefully everybody would \Nagree that that definitely is 0.
Dialogue: 0,0:01:51.54,0:01:56.66,Default,,0000,0000,0000,,Since this is an alternating series \Nand these two conditions have been met,
Dialogue: 0,0:01:56.66,0:02:00.02,Default,,0000,0000,0000,,that implies that this series right here,
Dialogue: 0,0:02:00.02,0:02:05.41,Default,,0000,0000,0000,,just like we drew out the diagram \Nof in the first video, converges.
Dialogue: 0,0:02:05.58,0:02:09.59,Default,,0000,0000,0000,,This series, n goes from 1 to infinity,
Dialogue: 0,0:02:09.59,0:02:14.79,Default,,0000,0000,0000,,negative 1 to the n minus 1 \Ndivided by n converges,
Dialogue: 0,0:02:14.79,0:02:19.33,Default,,0000,0000,0000,,and it converges by\Nthe alternating series test.
Dialogue: 0,0:02:22.89,0:02:26.01,Default,,0000,0000,0000,,We've got another alternating series here.
Dialogue: 0,0:02:26.09,0:02:28.38,Default,,0000,0000,0000,,This one starts with a negative term,
Dialogue: 0,0:02:28.38,0:02:31.69,Default,,0000,0000,0000,,but the formula that we have \Nis a little bit different.
Dialogue: 0,0:02:31.69,0:02:34.42,Default,,0000,0000,0000,,You can see I've listed out \Nthe first few terms.
Dialogue: 0,0:02:34.42,0:02:36.54,Default,,0000,0000,0000,,I've chosen not to reduce all the fractions
Dialogue: 0,0:02:36.54,0:02:39.89,Default,,0000,0000,0000,,just so that we can see the pattern \Nthat we've got going on here,
Dialogue: 0,0:02:39.89,0:02:42.86,Default,,0000,0000,0000,,and we're going to use the \Nalternating series test
Dialogue: 0,0:02:42.86,0:02:45.89,Default,,0000,0000,0000,,to try to determine whether \Nthis series converges or not.
Dialogue: 0,0:02:46.09,0:02:51.79,Default,,0000,0000,0000,,To begin with, \Nlet's figure out what b-n would be.
Dialogue: 0,0:02:51.79,0:02:54.29,Default,,0000,0000,0000,,That's the absolute value of each term.
Dialogue: 0,0:02:54.29,0:02:57.99,Default,,0000,0000,0000,,Basically, the only thing that affects \Nthe sign here is this part.
Dialogue: 0,0:02:57.99,0:03:03.79,Default,,0000,0000,0000,,That means the b-n would just \Nbe 3n divided by 4n plus 1.
Dialogue: 0,0:03:05.25,0:03:07.12,Default,,0000,0000,0000,,Now, it's not immediately obvious
Dialogue: 0,0:03:07.12,0:03:12.45,Default,,0000,0000,0000,,if these terms are actually \Nshrinking as n goes to infinity,
Dialogue: 0,0:03:12.100,0:03:15.45,Default,,0000,0000,0000,,we could look at the first few \Nand try to figure out
Dialogue: 0,0:03:15.45,0:03:17.83,Default,,0000,0000,0000,,whether those fractions \Nare getting smaller or not,
Dialogue: 0,0:03:17.83,0:03:22.85,Default,,0000,0000,0000,,but I would actually suggest ignoring \Nstep number 1 for the time being
Dialogue: 0,0:03:22.85,0:03:26.28,Default,,0000,0000,0000,,(just because that's a tougher \Nquestion to answer),
Dialogue: 0,0:03:26.28,0:03:28.48,Default,,0000,0000,0000,,and let's look at step 2.
Dialogue: 0,0:03:28.48,0:03:35.100,Default,,0000,0000,0000,,Let's try to figure out if the limit \Nas n goes to infinity for b-n is equal to 0.
Dialogue: 0,0:03:35.100,0:03:39.71,Default,,0000,0000,0000,,So if we actually write\Nin the formula for b-n,
Dialogue: 0,0:03:39.71,0:03:45.26,Default,,0000,0000,0000,,we're going to wind up with 3n,\Ndivided by 4n plus 1.
Dialogue: 0,0:03:45.26,0:03:49.84,Default,,0000,0000,0000,,And to do this limit, we can just divide \Neverything by the highest power of n,
Dialogue: 0,0:03:49.84,0:03:53.68,Default,,0000,0000,0000,,which is actually just n to the 1st.
Dialogue: 0,0:04:01.50,0:04:05.74,Default,,0000,0000,0000,,Now, of course, the n’s are going \Nto cancel in the first two terms,
Dialogue: 0,0:04:05.74,0:04:11.45,Default,,0000,0000,0000,,but this last term is going to wind up \Napproaching 0 as n goes to infinity,
Dialogue: 0,0:04:11.45,0:04:17.59,Default,,0000,0000,0000,,so we're going to be left with \N3 over 4 plus 0, or three fourths [3/4],
Dialogue: 0,0:04:17.59,0:04:20.19,Default,,0000,0000,0000,,and that is clearly not equal to 0.
Dialogue: 0,0:04:22.93,0:04:25.31,Default,,0000,0000,0000,,Since we failed this second condition,
Dialogue: 0,0:04:25.31,0:04:29.78,Default,,0000,0000,0000,,that actually means that the \Nalternating series test doesn't apply,
Dialogue: 0,0:04:29.78,0:04:32.36,Default,,0000,0000,0000,,so we may as well not \Neven try to figure out
Dialogue: 0,0:04:32.36,0:04:36.21,Default,,0000,0000,0000,,whether that first condition is met or not.
Dialogue: 0,0:04:36.21,0:04:39.49,Default,,0000,0000,0000,,But how do we determine whether \Nthe series converges or not?
Dialogue: 0,0:04:39.55,0:04:44.18,Default,,0000,0000,0000,,Well, fortunately, back in Section 11.2,
Dialogue: 0,0:04:44.18,0:04:47.72,Default,,0000,0000,0000,,we found out about something \Ncalled the test for divergence,
Dialogue: 0,0:04:47.72,0:04:54.48,Default,,0000,0000,0000,,and what that says is, if these terms \Nright here of the original series
Dialogue: 0,0:04:54.48,0:04:59.98,Default,,0000,0000,0000,,do not approach 0, then that means \Nthat the series would be divergent.
Dialogue: 0,0:04:59.98,0:05:07.62,Default,,0000,0000,0000,,If we were to look at the limit as n \Ngoes to infinity of the original terms,
Dialogue: 0,0:05:07.62,0:05:13.46,Default,,0000,0000,0000,,(negative 1 to the n, \Ntimes 3n, divided by 4 plus 1),
Dialogue: 0,0:05:13.46,0:05:15.62,Default,,0000,0000,0000,,what we would wind up finding out
Dialogue: 0,0:05:15.62,0:05:20.48,Default,,0000,0000,0000,,is that the absolute value\Nof the terms approach 3/4.
Dialogue: 0,0:05:20.98,0:05:24.03,Default,,0000,0000,0000,,But because of this \Nalternating portion here,
Dialogue: 0,0:05:24.03,0:05:26.23,Default,,0000,0000,0000,,that means that for large values of n,
Dialogue: 0,0:05:26.23,0:05:29.56,Default,,0000,0000,0000,,we're going to be approaching \Nnumbers that are close to positive 3/4
Dialogue: 0,0:05:29.56,0:05:31.45,Default,,0000,0000,0000,,and then negative 3/4,
Dialogue: 0,0:05:31.45,0:05:34.78,Default,,0000,0000,0000,,and then positive 3/4, \Nand then negative 3/4.
Dialogue: 0,0:05:34.78,0:05:39.94,Default,,0000,0000,0000,,And since that means the terms are not \Nactually coalescing around a single value,
Dialogue: 0,0:05:40.36,0:05:42.97,Default,,0000,0000,0000,,what does that tell us about this limit?
Dialogue: 0,0:05:42.97,0:05:47.61,Default,,0000,0000,0000,,Well, what that tells us \Nis that this limit does not exist.
Dialogue: 0,0:05:47.61,0:05:51.38,Default,,0000,0000,0000,,And if we look back at \Nthe test for divergence,
Dialogue: 0,0:05:51.38,0:05:55.58,Default,,0000,0000,0000,,it says that if the terms \Napproach any limit other than 0
Dialogue: 0,0:05:55.58,0:05:58.76,Default,,0000,0000,0000,,or if the limit of the terms does not exist,
Dialogue: 0,0:05:58.76,0:06:02.64,Default,,0000,0000,0000,,that means that the series \Nis going to be divergent.
Dialogue: 0,0:06:02.64,0:06:06.11,Default,,0000,0000,0000,,Therefore, by the test for divergence,
Dialogue: 0,0:06:06.11,0:06:09.74,Default,,0000,0000,0000,,it's actually not the alternating \Nseries test that tells us this result;
Dialogue: 0,0:06:09.74,0:06:12.53,Default,,0000,0000,0000,,it's actually the test for divergence.
Dialogue: 0,0:06:12.53,0:06:20.02,Default,,0000,0000,0000,,Because of that, we can say \Nthat this series has to diverge
Dialogue: 0,0:06:20.02,0:06:23.32,Default,,0000,0000,0000,,because if the individual \Nterms don't approach 0,
Dialogue: 0,0:06:23.32,0:06:25.80,Default,,0000,0000,0000,,then the series automatically diverges.\N