WEBVTT
1
00:00:01.139 --> 00:00:03.580 A:middle L:90%
Hello. So here we have these somewhere we have
2
00:00:03.589 --> 00:00:06.799 A:middle L:90%
and going from one to infinity of and over and
3
00:00:06.799 --> 00:00:10.179 A:middle L:90%
squared plus one. So to use the integral test
4
00:00:10.189 --> 00:00:12.210 A:middle L:90%
when we look at the function F of X is
5
00:00:12.220 --> 00:00:15.490 A:middle L:90%
equal to X over X squared plus one. And
6
00:00:15.490 --> 00:00:16.980 A:middle L:90%
we have to have the F of X is going
7
00:00:16.980 --> 00:00:20.379 A:middle L:90%
to be continuous, positive and decreasing. Um on
8
00:00:20.379 --> 00:00:24.559 A:middle L:90%
the interval from one to infinity. Well um f
9
00:00:24.559 --> 00:00:28.780 A:middle L:90%
of X is certainly continuous and positive um on one
10
00:00:28.780 --> 00:00:31.190 A:middle L:90%
to infinity. So then if we look at to
11
00:00:31.190 --> 00:00:34.179 A:middle L:90%
determine if um we are decreasing, we look at
12
00:00:34.179 --> 00:00:38.079 A:middle L:90%
the derivative. So look at fx again and take
13
00:00:38.079 --> 00:00:41.619 A:middle L:90%
the derivative F prime of X. So by the
14
00:00:41.619 --> 00:00:43.649 A:middle L:90%
quotient rule we have the F prime of X is
15
00:00:43.659 --> 00:00:46.659 A:middle L:90%
equal to one minus X squared divided by X squared
16
00:00:46.659 --> 00:00:51.869 A:middle L:90%
plus one quantity squared. And we see here that
17
00:00:51.880 --> 00:00:54.600 A:middle L:90%
well, for all x in one to infinity.
18
00:00:54.609 --> 00:00:57.869 A:middle L:90%
Um Yes, F prime of X is going to
19
00:00:57.869 --> 00:01:00.789 A:middle L:90%
be less than zero because one minus X squared is
20
00:01:00.799 --> 00:01:03.519 A:middle L:90%
less than zero for all X. Um in the
21
00:01:03.530 --> 00:01:07.030 A:middle L:90%
interval from one to infinity. So therefore the function
22
00:01:07.040 --> 00:01:11.469 A:middle L:90%
F of X. The function F of X which
23
00:01:11.469 --> 00:01:14.140 A:middle L:90%
is going to be equal to well X over X
24
00:01:14.140 --> 00:01:19.400 A:middle L:90%
squared plus one um is decreasing on the interval from
25
00:01:19.400 --> 00:01:23.480 A:middle L:90%
one to infinity. So therefore by the integral test
26
00:01:23.489 --> 00:01:29.930 A:middle L:90%
we then look at the integral from one to infinity
27
00:01:29.939 --> 00:01:33.819 A:middle L:90%
of F of X dx. So that's the integral
28
00:01:33.819 --> 00:01:36.689 A:middle L:90%
here from one to infinity of X over X squared
29
00:01:36.689 --> 00:01:38.640 A:middle L:90%
plus one dx. Which we can then right as
30
00:01:38.640 --> 00:01:42.659 A:middle L:90%
well, one half times the integral from one to
31
00:01:42.659 --> 00:01:48.430 A:middle L:90%
infinity of two. X over X squared plus one
32
00:01:48.439 --> 00:01:52.560 A:middle L:90%
D X. Um And then well this is gonna
33
00:01:52.560 --> 00:01:53.760 A:middle L:90%
then going to be equal to is ready like this
34
00:01:53.769 --> 00:01:57.579 A:middle L:90%
. This is then going to be equal to well
35
00:01:57.579 --> 00:02:04.299 A:middle L:90%
one half um times the limit as um let's say
36
00:02:04.310 --> 00:02:09.650 A:middle L:90%
T goes to infinity of well the natural log of
37
00:02:09.650 --> 00:02:15.169 A:middle L:90%
the absolute value of t squared plus one evaluated from
38
00:02:15.169 --> 00:02:17.719 A:middle L:90%
one to T. Uh And here we are.
39
00:02:17.719 --> 00:02:21.229 A:middle L:90%
This is going to be equal to well infinity.
40
00:02:21.240 --> 00:02:27.259 A:middle L:90%
So therefore we have that the integral from one to
41
00:02:27.259 --> 00:02:31.860 A:middle L:90%
infinity from one to infinity of F of X.
42
00:02:32.340 --> 00:02:38.810 A:middle L:90%
D X um is divergent. And therefore by the
43
00:02:38.819 --> 00:02:43.419 A:middle L:90%
integral test we can say that the somewhere end goes
44
00:02:43.419 --> 00:02:47.849 A:middle L:90%
from one to infinity of N over N squared plus
45
00:02:47.849 --> 00:02:57.379 A:middle L:90%
one must also be divergent. All right, take
46
00:02:57.379 --> A:middle L:90%
care.