Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated:
1
00:00:00,400 --> 00:00:04,830
In this video we're talking about how to use the limit comparison test to say whether a series converges
2
00:00:05,010 --> 00:00:06,240
or diverges.
3
00:00:06,240 --> 00:00:10,950
And in this particular problem we've been given the series from equals 1 to infinity of end squared
4
00:00:10,950 --> 00:00:13,890
divided by quantity and cute plus two.
5
00:00:13,890 --> 00:00:18,000
Now as a reminder the limit comparison test relies on a couple of conditions.
6
00:00:18,030 --> 00:00:23,790
First of all both the original series A7 and the comparison series be seben have to be greater than
7
00:00:24,030 --> 00:00:24,460
zero.
8
00:00:24,480 --> 00:00:29,760
So all the terms of the series have to be positive and in addition to that the limit as and goes to
9
00:00:29,760 --> 00:00:33,160
infinity of a sideband divided by B-7.
10
00:00:33,160 --> 00:00:37,590
So we're literally going to take the original series and divide it by the comparison series.
11
00:00:37,590 --> 00:00:43,020
If that limit the value of that limit is greater than zero then we can draw conclusions.
12
00:00:43,110 --> 00:00:49,510
We can say if the comparison series B-7 converges then the original series a seben also converges.
13
00:00:49,710 --> 00:00:55,960
Or if the comparison series B-7 diverges than the original series A7 also diverges.
14
00:00:56,100 --> 00:01:02,040
So we're going to first do is make sure that our original series and our comparison series meet these
15
00:01:02,040 --> 00:01:05,840
conditions before we go on to drawing any conclusion.
16
00:01:05,850 --> 00:01:08,780
So what will be our comparison series.
17
00:01:08,880 --> 00:01:14,370
Well what we usually want to do if we have a rational function or a fraction is we want to take the
18
00:01:14,370 --> 00:01:19,710
term from the numerator and denominator that's going to have the greatest effect on the numerator and
19
00:01:19,710 --> 00:01:22,510
denominator or the term with the greatest magnitude.
20
00:01:22,650 --> 00:01:26,820
So in this case there's only one term in the numerator it's end squared.
21
00:01:26,820 --> 00:01:32,520
So we're going to go ahead and take and squared for the new numerator and then for the denominator and
22
00:01:32,520 --> 00:01:37,390
cubed has a greater magnitude or greater degree than just the constant 2.
23
00:01:37,410 --> 00:01:41,010
So we're going to take and cubed only for the denominator.
24
00:01:41,010 --> 00:01:45,870
This is going to simplify we're going to be able to cancel and squared from both the numerator and denominator
25
00:01:46,230 --> 00:01:52,410
and this will simplify to 1 over n that means we're going to call our comparison series 1 over and so
26
00:01:52,410 --> 00:01:59,310
let's go ahead and say that this series from N equals 1 to infinity of 1 over n and we'll go ahead and
27
00:01:59,310 --> 00:02:05,680
call the original series a 7 and we'll call the comparison Series B 7.
28
00:02:06,030 --> 00:02:10,510
So now we want to do is show that both of these series are always going to be positive.
29
00:02:10,530 --> 00:02:15,570
So if we look at the original series we know we're starting at and equals 1 and we're counting up and
30
00:02:15,570 --> 00:02:17,730
equals 2 equals 3 and equals four.
31
00:02:17,850 --> 00:02:19,580
All the way up to infinity.
32
00:02:19,700 --> 00:02:24,440
Well if we plug in N equals 1 in the numerator here we can get 1 squared which is 1.
33
00:02:24,480 --> 00:02:29,430
Here we're going to get one plus two is three so we're going to end up with one third.
34
00:02:29,610 --> 00:02:35,490
If we plug in and equals 2 we're going to get 4 over 8 plus 2 is 10 for over 10.
35
00:02:35,490 --> 00:02:37,920
Or two over five.
36
00:02:38,250 --> 00:02:43,540
If we plug in N equals three we're going to get eight over twenty seven plus two is 29.
37
00:02:43,590 --> 00:02:47,640
So we're going to get 8 over 29 and we could keep going.
38
00:02:47,640 --> 00:02:51,040
Let's go ahead and look at though the comparison Series B 7.
39
00:02:51,150 --> 00:02:59,610
We're also starting at any one equals one we get 1 over 1 or 1 at N equals 2 we get 1 1/2 and equals
40
00:02:59,610 --> 00:03:02,690
3 we get one third and we could see that we could keep going.
41
00:03:02,700 --> 00:03:04,910
But in either case there's no value of.
42
00:03:04,920 --> 00:03:10,530
And then we could plug in if we're counting up from N equals 1 2 3 4 et cetera that we could plug into
43
00:03:10,530 --> 00:03:14,190
either series that would make any of these terms negative.
44
00:03:14,250 --> 00:03:20,220
And so we can say then that a 7 is always going to be greater than zero and we can say that B-7 is always
45
00:03:20,220 --> 00:03:21,670
going to be greater than zero.
46
00:03:21,930 --> 00:03:26,160
So those two conditions have been met and the next thing we want to show is that the limit is and goes
47
00:03:26,160 --> 00:03:30,670
to infinity of the ratio of these series is going to be greater than zero.
48
00:03:30,690 --> 00:03:38,160
So if we take the limit as and approaches infinity of a sob in our original series and squared over
49
00:03:38,580 --> 00:03:44,310
and cubed plus two and we divide it by our comparison series we're going to divide this whole thing
50
00:03:44,580 --> 00:03:45,800
by one over and.
51
00:03:45,810 --> 00:03:51,300
But keep in mind that that's exactly the same thing as multiplying by an over 1.
52
00:03:51,300 --> 00:03:57,870
The result then here is going to be the limit as and goes to infinity will multiply and squared by and
53
00:03:57,920 --> 00:04:02,180
we'll get and cubed over and cubed plus two.
54
00:04:02,190 --> 00:04:07,140
Now in order to take the limit what we want to do is we want to multiply both the numerator and denominator
55
00:04:07,470 --> 00:04:14,390
by the highest degree variables so we have one over and cubed and one over and cubed like this.
56
00:04:14,390 --> 00:04:20,580
What that's going to do when we say one over and cubed times and cubed we're going to get one in the
57
00:04:20,680 --> 00:04:26,760
numerator so we're going to have limit as and approaches infinity we're going to have one in the numerator
58
00:04:27,210 --> 00:04:31,730
and denominator one over and cubed times and cubed is going to give us 1.
59
00:04:32,010 --> 00:04:38,610
Then we're going to have here plus two over and cubed that way when we take the limit as and goes to
60
00:04:38,610 --> 00:04:44,640
infinity the denominator right here is going to get extremely large to divided by an infinitely large
61
00:04:44,640 --> 00:04:46,490
number is going to be zero.
62
00:04:46,500 --> 00:04:50,670
So this term is going to go to zero which means it's going to become zero there.
63
00:04:50,880 --> 00:04:57,240
And we're just going to end up with one over 1 or that this limit here if we call this limit L that
64
00:04:57,330 --> 00:05:02,440
L is going to be equal to 1 and one is greater than zero.
65
00:05:02,460 --> 00:05:07,990
So we've shown that this condition here has also been met because Al is greater than zero.
66
00:05:08,100 --> 00:05:11,990
So now that those conditions have been met we can go on to our conclusions.
67
00:05:12,060 --> 00:05:17,290
What we want to do is evaluate be stubborn and say whether B-7 converges or diverges.
68
00:05:17,370 --> 00:05:23,130
If we look here at B-7 we notice that it's the harmonic series one divided by and is a harmonic series
69
00:05:23,490 --> 00:05:26,640
and we know that the harmonic series diverges.
70
00:05:26,880 --> 00:05:31,790
So since B Subban diverges we're looking at this conclusion right here.
71
00:05:31,890 --> 00:05:35,570
If B-7 diverges then A7 also diverges.
72
00:05:35,670 --> 00:05:44,790
So since we've shown that B Subban diverges we can conclude that a Sabun also diverges and that's how
73
00:05:44,790 --> 00:05:49,800
you use the limit comparison test to say whether a series converges or diverges.
8657
Can't find what you're looking for?
Get subtitles in any language from opensubtitles.com, and translate them here.