1
00:00:00,340 --> 00:00:01,855
N, log N.

2
00:00:01,855 --> 00:00:02,595
[SOUND]

3
00:00:02,595 --> 00:00:10,291
The harmonic series diverges, but the sum
of the reciprocals

4
00:00:10,291 --> 00:00:16,950
of the squares converges.
P series with P equals 2 converges.

5
00:00:16,950 --> 00:00:18,910
Alright.
So this series converges.

6
00:00:18,910 --> 00:00:23,180
The harmonic series diverges, so this
series diverges.

7
00:00:23,180 --> 00:00:24,850
But what's something in between, right?

8
00:00:24,850 --> 00:00:29,210
Where's the boundary between convergence
and divergence?

9
00:00:29,210 --> 00:00:32,470
Well here's an example of a series that
sits in between.

10
00:00:32,470 --> 00:00:35,740
Its the sum N goes from 4 to infinity of 1
over N log N.

11
00:00:36,810 --> 00:00:39,330
Now, 1 over N log N is less than 1 over

12
00:00:39,330 --> 00:00:42,080
N, because N times log N is bigger than N,
right?

13
00:00:42,080 --> 00:00:44,400
A bigger denominator makes the fraction
smaller.

14
00:00:45,480 --> 00:00:48,810
And because N squared is bigger than N log
N, 1

15
00:00:48,810 --> 00:00:52,170
over N squared is less than 1 over N log
N.

16
00:00:52,170 --> 00:00:54,370
So, this thing is bigger

17
00:00:54,370 --> 00:00:56,640
than something that converges, smaller
than something that

18
00:00:56,640 --> 00:01:00,720
diverges, so it's, it's really a question,
right?

19
00:01:00,720 --> 00:01:04,380
Does this series converge or diverge?

20
00:01:04,380 --> 00:01:06,590
We can use Cauchy Condensation.

21
00:01:06,590 --> 00:01:09,760
So let's analyze this series using Cauchy
Condensation.

22
00:01:09,760 --> 00:01:11,580
So I'll write down the condensed series.

23
00:01:11,580 --> 00:01:18,430
It's the sum N goes from 2 to infinity of
2 to the N times the 2 to the Nth term.

24
00:01:18,430 --> 00:01:19,590
So 2 to the Nth term

25
00:01:19,590 --> 00:01:25,240
mean I put a 2 to the N times.
Log 2 to the N in the denominator.

26
00:01:25,240 --> 00:01:27,810
Okay, so this is the condensed series,
this is the original series.

27
00:01:27,810 --> 00:01:32,660
Cauchy Condensation tells me that these
two series share the same fate.

28
00:01:32,660 --> 00:01:35,310
They either both converge or both diverge.

29
00:01:35,310 --> 00:01:37,680
So it's enough to figure out what's going
on with this series.

30
00:01:37,680 --> 00:01:38,860
Oh and look!

31
00:01:38,860 --> 00:01:41,136
I've got a 2 to the N in the numerator and
a 2 to

32
00:01:41,136 --> 00:01:45,450
the N in the denominator, so we cancel
those and we can rewrite this as

33
00:01:45,450 --> 00:01:52,140
the sum N goes from 2 to infinity, which
is 1 over log 2 to the N.

34
00:01:53,890 --> 00:01:56,500
Now I could use property of logs to
simplify this.

35
00:01:56,500 --> 00:02:03,540
This is the sum N goes from 2 to infinity
of 1 over N times log 2.

36
00:02:03,540 --> 00:02:06,753
That's just because log 2 to the N is N
times log 2.

37
00:02:07,870 --> 00:02:09,560
And then, well look at this.

38
00:02:09,560 --> 00:02:11,480
This is

39
00:02:11,480 --> 00:02:18,290
1 over log 2 times the sum N goes from 2
infinity of 1 over N.

40
00:02:18,290 --> 00:02:21,770
Oh, but this is bad or really great news
depending on your attitude.

41
00:02:21,770 --> 00:02:25,260
This thing here, is the tail of a harmonic
series.

42
00:02:25,260 --> 00:02:30,790
So that means that the condensed series
diverges.

43
00:02:30,790 --> 00:02:36,550
And because the condensed series diverges,
this original series must also diverge.

44
00:02:36,550 --> 00:02:39,140
If you like integrals, you can use an
integral test.

45
00:02:39,140 --> 00:02:39,490
Okay.

46
00:02:39,490 --> 00:02:41,930
So let's do this with the integral test.

47
00:02:41,930 --> 00:02:44,460
So I could look at this integral, the
integral from

48
00:02:44,460 --> 00:02:49,360
4 to infinity of 1 over X times log X DX.

49
00:02:49,360 --> 00:02:51,500
That's a suitable function to consider.

50
00:02:51,500 --> 00:02:56,320
By definition, this integral from 4 to
infinity just means the limit as N

51
00:02:56,320 --> 00:03:01,640
approaches infinity of the integral from 4
to big N of 1 over

52
00:03:01,640 --> 00:03:04,240
X times log X DX.

53
00:03:04,240 --> 00:03:06,600
But now, how do I evaluate that definite
integral?

54
00:03:07,740 --> 00:03:09,170
Well, I happen to know this.

55
00:03:09,170 --> 00:03:14,815
The derivative of log of log of X is what?

56
00:03:14,815 --> 00:03:16,170
Well by the chain rule, this is the

57
00:03:16,170 --> 00:03:19,650
derivative log, which is 1 over, evaluate
the

58
00:03:19,650 --> 00:03:25,000
inside, log of X, times the derivative of
the inside function, which is 1 over X.

59
00:03:25,000 --> 00:03:26,750
Well, here I've got 1 over

60
00:03:26,750 --> 00:03:30,480
X times log X.
Here, I've got some other function.

61
00:03:30,480 --> 00:03:33,210
Here I've got an anti-derivative for this
integrand.

62
00:03:33,210 --> 00:03:35,560
So, that's enough for me to be able to

63
00:03:35,560 --> 00:03:40,090
calculate this definite integral using the
fundamental theorem of calculus.

64
00:03:40,090 --> 00:03:46,570
So this is the limit as N approaches
infinity of what is this?

65
00:03:46,570 --> 00:03:52,850
This is telling me an anti-derivative, so
log log X evaluated

66
00:03:52,850 --> 00:03:53,790
at 4 and N.

67
00:03:53,790 --> 00:03:58,600
And then I could plug in N and plug in 4
and take the difference.

68
00:03:58,600 --> 00:04:06,510
This is the limit N goes to infinity of
log log N minus log log 4.

69
00:04:06,510 --> 00:04:09,970
What is this limit?

70
00:04:09,970 --> 00:04:14,510
Well, it's growing very slowly but it is,
in fact, running off to infinity, right.

71
00:04:14,510 --> 00:04:17,820
By choosing N big enough, I can make log N
as large as I like,

72
00:04:17,820 --> 00:04:21,840
and that means I can also make log log N
as large as I like.

73
00:04:21,840 --> 00:04:27,090
So this limit is in fact, infinity.

74
00:04:27,090 --> 00:04:30,670
That means that this integral diverges.

75
00:04:30,670 --> 00:04:34,530
That means that the original series
diverges as well.

76
00:04:34,530 --> 00:04:36,690
So that diverges, but maybe if we mess
around

77
00:04:36,690 --> 00:04:39,300
with it a bit we can make it converge.

78
00:04:39,300 --> 00:04:42,130
So our original question was whether or
not this series

79
00:04:42,130 --> 00:04:43,660
converged or diverged.

80
00:04:43,660 --> 00:04:46,850
And now we've seen, both by using
condensation

81
00:04:46,850 --> 00:04:50,870
and the integral test, that this series
diverges.

82
00:04:50,870 --> 00:04:52,630
We're going to go back to our original
story, right?

83
00:04:52,630 --> 00:04:57,660
We were thinking of this series as somehow
sitting in between these two series.

84
00:04:58,930 --> 00:05:02,550
But now where's the boundary between
convergence and divergence, right?

85
00:05:02,550 --> 00:05:06,270
This series converges, these two series
diverge.

86
00:05:06,270 --> 00:05:07,510
So let me try to fit another

87
00:05:07,510 --> 00:05:11,760
series in between these two series.
Here's an example.

88
00:05:11,760 --> 00:05:14,360
Its the sum N goes from 4 to infinity of 1

89
00:05:14,360 --> 00:05:18,570
over N times log N, and that log N is
squared.

90
00:05:18,570 --> 00:05:21,660
Does that series converge or diverge?

91
00:05:22,980 --> 00:05:27,050
We can use condensation.
So lets write down the condensed series.

92
00:05:27,050 --> 00:05:32,630
In this case, this is the sum N goes from
2 to infinity of 2

93
00:05:32,630 --> 00:05:36,050
to the N times the 2 to the Nth term.

94
00:05:36,050 --> 00:05:41,120
So, 2 to the N times log 2 to the N
squared.

95
00:05:42,230 --> 00:05:44,340
Now, how do I evaluate this series, right?

96
00:05:44,340 --> 00:05:46,360
If I can determine the convergence of this

97
00:05:46,360 --> 00:05:49,440
series, I've determined the convergence of
the original series.

98
00:05:49,440 --> 00:05:50,190
Good news.

99
00:05:50,190 --> 00:05:52,180
This 2 to the N and this 2 to the N
cancel.

100
00:05:52,180 --> 00:05:57,680
So now I'm left with the sum N goes from 2
to

101
00:05:57,680 --> 00:06:01,510
infinity of 1 over log 2 to

102
00:06:01,510 --> 00:06:05,150
the N squared.
What else can I do there?

103
00:06:05,150 --> 00:06:11,205
Oh, I can use properties of logs again.
So this is the sum N goes

104
00:06:11,205 --> 00:06:17,640
from 2 to infinity of 1 over N times log 2
squared.

105
00:06:19,090 --> 00:06:23,170
I could factor out the 1 over log 2
squared.

106
00:06:23,170 --> 00:06:28,874
So this is 1 over log 2 squared times the
sum N goes from

107
00:06:28,874 --> 00:06:34,730
2 to infinity of 1 over N squared.
this is really great.

108
00:06:34,730 --> 00:06:35,080
Right?

109
00:06:35,080 --> 00:06:39,190
because what do I know?
This is a P series, where P equals 2.

110
00:06:39,190 --> 00:06:40,700
This series converges.

111
00:06:41,700 --> 00:06:44,270
Well, that means that this condense series
converges.

112
00:06:44,270 --> 00:06:48,640
That means the original series converges.
So, we've got a convergence

113
00:06:48,640 --> 00:06:52,689
series and a divergence series.
It's worth comparing these two series.

114
00:06:53,780 --> 00:06:55,830
In this series, which we were asking

115
00:06:55,830 --> 00:06:59,180
about, we now know converges by Cauchy
condensation.

116
00:06:59,180 --> 00:07:05,090
And we've already seen a little while ago
that this series diverges.

117
00:07:05,090 --> 00:07:06,650
So again, we could try to play the same
game, there's

118
00:07:06,650 --> 00:07:09,240
just something that we try to fit in
between these two.

119
00:07:10,240 --> 00:07:11,670
Well, here's our question.

120
00:07:11,670 --> 00:07:13,778
Does this series, which sort of fits

121
00:07:13,778 --> 00:07:18,110
in between these two series, does this
series converge or diverge?

122
00:07:18,110 --> 00:07:21,638
This is the series N goes from 4 to
infinity

123
00:07:21,638 --> 00:07:24,991
over 1 over N times log N times log log N.

124
00:07:24,991 --> 00:07:27,740
Well, I'll leave it to you to analyze this
series.

125
00:07:27,740 --> 00:07:37,740
[SOUND]