WEBVTT

00:00:00.015 --> 00:00:03.724
We're going to try to determine
whether this alternating series

00:00:03.724 --> 00:00:08.189
converges or diverges
using the alternating series test.

00:00:08.189 --> 00:00:12.107
The first thing we have to figure out 
is actually a formula for b-n.

00:00:12.107 --> 00:00:18.072
If you recall, b-n is basically the 
absolute value of each of the terms.

00:00:18.072 --> 00:00:19.457
And if we're looking at this,

00:00:19.457 --> 00:00:22.939
we just ignore the part 
that causes the sign to alternate,

00:00:22.939 --> 00:00:28.221
Hopefully you would agree
that b-n is just 1 over n.

00:00:28.221 --> 00:00:29.573
The first term is 1,

00:00:29.573 --> 00:00:32.490
the second term we're 
subtracting is one half [1/2],

00:00:32.490 --> 00:00:33.605
and then one third [1/3],

00:00:33.605 --> 00:00:37.139
then we subtract one fourth [1/4], 
and we add one fifth [1/5].

00:00:37.139 --> 00:00:40.354
That would be our expression for b-n.

00:00:40.354 --> 00:00:45.871
There are two conditions 
that need to be met for this test.

00:00:45.871 --> 00:00:51.391
So the first condition is 
that the n plus first term

00:00:51.391 --> 00:00:55.372
is supposed to be less than 
or equal to the nth term

00:00:55.372 --> 00:00:58.858
for all values of n beyond a certain point.

00:00:58.858 --> 00:01:04.507
And to test that, we just have to figure 
out what the n plus first term would be;

00:01:04.507 --> 00:01:08.305
of course, that would be 1 over n plus 1.

00:01:08.305 --> 00:01:11.909
If we compare that to the nth term, 
which is 1 over n,

00:01:11.909 --> 00:01:17.440
clearly, 1 over n plus 1 
is less than 1 over n,

00:01:17.440 --> 00:01:20.491
so that satisfies the first condition.

00:01:20.491 --> 00:01:30.555
The second condition is that the limit as 
n goes to infinity for b-n needs to equal 0,

00:01:30.555 --> 00:01:32.422
so that's the next thing to test.

00:01:32.422 --> 00:01:36.423
And in some examples, 
we'll actually do this first

00:01:36.423 --> 00:01:40.657
because if this is not true, 
then the whole test is going to fail.

00:01:40.657 --> 00:01:47.073
But in this case, if we look at the limit 
as n goes to infinity for 1 over n,

00:01:47.073 --> 00:01:51.540
hopefully everybody would 
agree that that definitely is 0.

00:01:51.540 --> 00:01:56.658
Since this is an alternating series 
and these two conditions have been met,

00:01:56.658 --> 00:02:00.024
that implies that this series right here,

00:02:00.024 --> 00:02:05.407
just like we drew out the diagram 
of in the first video, converges.

00:02:05.576 --> 00:02:09.591
This series, n goes from 1 to infinity,

00:02:09.591 --> 00:02:14.792
negative 1 to the n minus 1 
divided by n converges,

00:02:14.792 --> 00:02:19.334
and it converges by
the alternating series test.

00:02:22.893 --> 00:02:26.006
We've got another alternating series here.

00:02:26.091 --> 00:02:28.375
This one starts with a negative term,

00:02:28.375 --> 00:02:31.689
but the formula that we have 
is a little bit different.

00:02:31.689 --> 00:02:34.424
You can see I've listed out 
the first few terms.

00:02:34.424 --> 00:02:36.540
I've chosen not to reduce all the fractions

00:02:36.540 --> 00:02:39.890
just so that we can see the pattern 
that we've got going on here,

00:02:39.890 --> 00:02:42.858
and we're going to use the 
alternating series test

00:02:42.858 --> 00:02:45.890
to try to determine whether 
this series converges or not.

00:02:46.088 --> 00:02:51.787
To begin with, 
let's figure out what b-n would be.

00:02:51.787 --> 00:02:54.289
That's the absolute value of each term.

00:02:54.289 --> 00:02:57.988
Basically, the only thing that affects 
the sign here is this part.

00:02:57.988 --> 00:03:03.787
That means the b-n would just 
be 3n divided by 4n plus 1.

00:03:05.254 --> 00:03:07.120
Now, it's not immediately obvious

00:03:07.120 --> 00:03:12.454
if these terms are actually 
shrinking as n goes to infinity,

00:03:12.996 --> 00:03:15.446
we could look at the first few 
and try to figure out

00:03:15.446 --> 00:03:17.833
whether those fractions 
are getting smaller or not,

00:03:17.833 --> 00:03:22.852
but I would actually suggest ignoring 
step number 1 for the time being

00:03:22.852 --> 00:03:26.279
(just because that's a tougher 
question to answer),

00:03:26.279 --> 00:03:28.481
and let's look at step 2.

00:03:28.481 --> 00:03:35.996
Let's try to figure out if the limit 
as n goes to infinity for b-n is equal to 0.

00:03:35.996 --> 00:03:39.712
So if we actually write
in the formula for b-n,

00:03:39.712 --> 00:03:45.263
we're going to wind up with 3n,
divided by 4n plus 1.

00:03:45.263 --> 00:03:49.837
And to do this limit, we can just divide 
everything by the highest power of n,

00:03:49.837 --> 00:03:53.681
which is actually just n to the 1st.

00:04:01.501 --> 00:04:05.739
Now, of course, the n’s are going 
to cancel in the first two terms,

00:04:05.739 --> 00:04:11.450
but this last term is going to wind up 
approaching 0 as n goes to infinity,

00:04:11.450 --> 00:04:17.586
so we're going to be left with 
3 over 4 plus 0, or three fourths [3/4],

00:04:17.586 --> 00:04:20.186
and that is clearly not equal to 0.

00:04:22.926 --> 00:04:25.310
Since we failed this second condition,

00:04:25.310 --> 00:04:29.777
that actually means that the 
alternating series test doesn't apply,

00:04:29.777 --> 00:04:32.361
so we may as well not 
even try to figure out

00:04:32.361 --> 00:04:36.213
whether that first condition is met or not.

00:04:36.213 --> 00:04:39.494
But how do we determine whether 
the series converges or not?

00:04:39.553 --> 00:04:44.181
Well, fortunately, back in Section 11.2,

00:04:44.181 --> 00:04:47.718
we found out about something 
called the test for divergence,

00:04:47.718 --> 00:04:54.481
and what that says is, if these terms 
right here of the original series

00:04:54.481 --> 00:04:59.981
do not approach 0, then that means 
that the series would be divergent.

00:04:59.981 --> 00:05:07.617
If we were to look at the limit as n 
goes to infinity of the original terms,

00:05:07.617 --> 00:05:13.465
(negative 1 to the n, 
times 3n, divided by 4 plus 1),

00:05:13.465 --> 00:05:15.617
what we would wind up finding out

00:05:15.617 --> 00:05:20.483
is that the absolute value
of the terms approach 3/4.

00:05:20.980 --> 00:05:24.030
But because of this 
alternating portion here,

00:05:24.030 --> 00:05:26.230
that means that for large values of n,

00:05:26.230 --> 00:05:29.562
we're going to be approaching 
numbers that are close to positive 3/4

00:05:29.562 --> 00:05:31.446
and then negative 3/4,

00:05:31.446 --> 00:05:34.781
and then positive 3/4, 
and then negative 3/4.

00:05:34.781 --> 00:05:39.945
And since that means the terms are not 
actually coalescing around a single value,

00:05:40.362 --> 00:05:42.974
what does that tell us about this limit?

00:05:42.974 --> 00:05:47.608
Well, what that tells us 
is that this limit does not exist.

00:05:47.608 --> 00:05:51.381
And if we look back at 
the test for divergence,

00:05:51.381 --> 00:05:55.575
it says that if the terms 
approach any limit other than 0

00:05:55.575 --> 00:05:58.758
or if the limit of the terms does not exist,

00:05:58.758 --> 00:06:02.640
that means that the series 
is going to be divergent.

00:06:02.640 --> 00:06:06.112
Therefore, by the test for divergence,

00:06:06.112 --> 00:06:09.741
it's actually not the alternating 
series test that tells us this result;

00:06:09.741 --> 00:06:12.526
it's actually the test for divergence.

00:06:12.526 --> 00:06:20.016
Because of that, we can say 
that this series has to diverge

00:06:20.016 --> 00:06:23.320
because if the individual 
terms don't approach 0,

00:06:23.320 --> 00:06:25.805
then the series automatically diverges.