1
00:00:00,260 --> 00:00:03,350
Let's sum the reciprocals of squares.

2
00:00:03,350 --> 00:00:09,670
[SOUND]

3
00:00:09,670 --> 00:00:14,320
There's a few different approaches to
trying to determine whether or not this

4
00:00:14,320 --> 00:00:19,450
series converges or diverges.
So here's the series I'm interested in.

5
00:00:19,450 --> 00:00:24,620
The sum, n goes from 1 to infinity, of 1
over n squared.

6
00:00:24,620 --> 00:00:30,220
And that series converges if and only if
the series

7
00:00:30,220 --> 00:00:36,250
n starts at 2 to infinity of 1over n
squared converges.

8
00:00:36,250 --> 00:00:38,400
All right, whether or not I include the
end equals

9
00:00:38,400 --> 00:00:41,480
1 term doesn't affect the convergence of
the series at all.

10
00:00:41,480 --> 00:00:44,699
So let's try to analyse this series now.

11
00:00:45,770 --> 00:00:49,200
Well, note that 0 is less than or equal to
1 over n

12
00:00:49,200 --> 00:00:55,490
squared is less than or equal to 1 over n
squared minus n.

13
00:00:55,490 --> 00:00:59,100
I mean, at least as long as n is 2 or
more.

14
00:00:59,100 --> 00:01:01,490
I don't want to plug in n equals 1 here,
because then I'd be dividing

15
00:01:01,490 --> 00:01:05,030
by 0.
But this is true because this denominator

16
00:01:05,030 --> 00:01:09,890
is smaller than this denominator so this
fraction is bigger than this fraction.

17
00:01:09,890 --> 00:01:13,890
Now what does that mean about the series.
Well i can do a comparison test then.

18
00:01:14,920 --> 00:01:19,550
So by comparison test.

19
00:01:21,470 --> 00:01:26,550
The sum of 1 over n squared minus n, n
goes from

20
00:01:26,550 --> 00:01:33,070
2 to infinity converges.
Implies that the smaller sum,

21
00:01:34,110 --> 00:01:40,310
the sum n goes from 2 to infinity of 1
over n squared convergences.

22
00:01:40,310 --> 00:01:44,400
So now I want to analyse the sum of 1 over
n squared minus n.

23
00:01:44,400 --> 00:01:50,420
Well, it's going to turn out that this
series telescopes.

24
00:01:50,420 --> 00:01:51,970
Let's see how that goes.

25
00:01:51,970 --> 00:01:57,360
So, the series that I'm interested in
analysing is, the sum

26
00:01:57,360 --> 00:02:02,085
n goes from 2 to infinity of 1 over n
squared minus n.

27
00:02:02,085 --> 00:02:04,730
Al right, I'd like to evaluate that
series.

28
00:02:04,730 --> 00:02:06,340
Now, how could I do that?

29
00:02:06,340 --> 00:02:12,200
The trick is to notice the following.
What's 1 over n minus 1 minus 1 over n?

30
00:02:14,350 --> 00:02:19,990
I can put that over a common denominator.
So this is n over n times n minus

31
00:02:19,990 --> 00:02:27,070
1 minus n minus 1 over n times n minus 1.

32
00:02:27,070 --> 00:02:29,930
And now that I've got it over a common
denominator, I can do the subtraction.

33
00:02:29,930 --> 00:02:35,780
This is n minus n minus 1 over n times n
minus 1.

34
00:02:35,780 --> 00:02:39,940
Well, n minus n minus 1, that's just 1,
and

35
00:02:39,940 --> 00:02:45,840
the denominator is n-squared minus n, n
squared minus n.

36
00:02:45,840 --> 00:02:50,800
So, evaluating this series is really the
same as evaluating

37
00:02:50,800 --> 00:02:55,530
1 over n minus 1, minus 1 over n, the sum
of these.

38
00:02:56,570 --> 00:03:00,080
And before plunging into the infinity on
top, let's

39
00:03:00,080 --> 00:03:03,450
just do this from 2 to some value big n.

40
00:03:04,760 --> 00:03:05,010
So what

41
00:03:05,010 --> 00:03:10,390
happens when I plug in 2, I get 1 over 2
minus 1 is 1 minus 1 over 2.

42
00:03:10,390 --> 00:03:17,560
And then I plug in n equals 3 I get 1 over
3 minus 1 which is 2 minus 1 over 3.

43
00:03:17,560 --> 00:03:22,200
And then I plug in n equals 4 and then I
get

44
00:03:22,200 --> 00:03:26,070
1 over 4 and I get 1 over 3 minus 1 over
4.

45
00:03:26,070 --> 00:03:30,430
And then I keep on going until I plug in n
and I get 1 over

46
00:03:30,430 --> 00:03:36,340
big N minus 1 minus 1 over n.
But lots and lots of stuff cancels.

47
00:03:36,340 --> 00:03:38,630
This is exactly what I mean by
telescoping, right?

48
00:03:38,630 --> 00:03:41,580
What cancels is minus and a 1 2 and this a
1 2, this minus a

49
00:03:41,580 --> 00:03:45,170
1 3 and this a 1 3, this minus a 1 4 will
cancel something in there.

50
00:03:45,170 --> 00:03:47,230
Everything else in the middle will die.

51
00:03:47,230 --> 00:03:50,030
There'll be a 1 over N minus 1 term, with

52
00:03:50,030 --> 00:03:52,440
a negative sign in front of it, which'll
cancel this.

53
00:03:52,440 --> 00:03:55,220
The only thing that survives is this
initial term,

54
00:03:55,220 --> 00:03:58,650
1 over 1, and this last term, minus 1 over
N.

55
00:04:00,700 --> 00:04:05,730
So the sum of this, little n from 2 to big
N, is this.

56
00:04:05,730 --> 00:04:08,430
Now how do I evaluate the infinite series?

57
00:04:08,430 --> 00:04:11,480
Well I just take a limit as big N goes to
infinity.

58
00:04:11,480 --> 00:04:20,030
So the limit as big N goes to infinity of
the sum little n goes from 2 to big N

59
00:04:20,030 --> 00:04:26,240
of 1 over little n minus 1 minus 1 over n.
Is the limit as

60
00:04:26,240 --> 00:04:31,780
big N goes to infinity of 1 over 1 minus 1
over big N.

61
00:04:31,780 --> 00:04:35,780
Well, as big N goes to infinity 1 over N
this term is going to 0.

62
00:04:35,780 --> 00:04:38,660
So, it's 1 over 1 minus something very
close to 0.

63
00:04:38,660 --> 00:04:40,930
This limit is 1.

64
00:04:40,930 --> 00:04:47,540
And that means, that this original series
not only converges but I know its value.

65
00:04:47,540 --> 00:04:49,200
Its value at 1.

66
00:04:49,200 --> 00:04:51,210
Why is that significant?
Well,

67
00:04:51,210 --> 00:04:58,980
knowing that this series converges, then
means that this series converges.

68
00:04:58,980 --> 00:05:03,610
And if this series, little n from 2 to
infinity of 1 over n squared converge.

69
00:05:03,610 --> 00:05:06,570
That means that the original series that
I'm interested in,

70
00:05:06,570 --> 00:05:10,820
the sum of the reciprocals of the squares,
that converges.

71
00:05:10,820 --> 00:05:12,180
And that's great.

72
00:05:12,180 --> 00:05:16,370
But that's not the only way to determine
that this series converges.

73
00:05:16,370 --> 00:05:20,310
So, yeah, I want to apply Cauchy
Condensation, but to what series?

74
00:05:20,310 --> 00:05:22,350
And I'm still working on this series, the
sum of

75
00:05:22,350 --> 00:05:25,460
1 over n squared, n goes from 1 to
infinity.

76
00:05:25,460 --> 00:05:28,670
But instead of looking at that series, I'm
going to write it like this.

77
00:05:28,670 --> 00:05:34,150
The sum of just the a sub n's where a sub
n is 1 over n squared.

78
00:05:34,150 --> 00:05:38,190
You know, I want to note, what, about this
sequence a sub n?

79
00:05:38,190 --> 00:05:41,840
It's a sequence who's terms are positive
and it's

80
00:05:41,840 --> 00:05:43,600
a decreasing sequence.

81
00:05:43,600 --> 00:05:46,950
So it's this sort of thing that I'm
allowed to apply Cauchy Condensation to.

82
00:05:46,950 --> 00:05:49,790
And what does Cauchy Condensation say?

83
00:05:49,790 --> 00:05:52,290
Well, it says that this series converges

84
00:05:52,290 --> 00:05:55,270
if and only if the condensed series
converges.

85
00:05:55,270 --> 00:05:58,570
If and only if the sum of 2 to the n times

86
00:05:58,570 --> 00:06:03,050
the 2 to the nth term, n goes from zero to
infinity, converges.

87
00:06:03,050 --> 00:06:04,990
So what's the condensed series?

88
00:06:04,990 --> 00:06:06,840
In this case, just asking

89
00:06:06,840 --> 00:06:12,750
whats this series in this case.
Well that's the sum n goes from

90
00:06:12,750 --> 00:06:18,940
0 to infinity of 2 to the n times what's
the 2 to the n term of this sequence?

91
00:06:18,940 --> 00:06:22,970
its 1 over instead of n, am going to write
2 to the n squared.

92
00:06:24,200 --> 00:06:28,820
But I can simplify this, this is something
times 1 over the same thing squared.

93
00:06:28,820 --> 00:06:32,180
Well that's the sum n goes from 0 to
infinity just

94
00:06:32,180 --> 00:06:38,250
1 over 2 to the n.
Does that series converge or diverge?

95
00:06:38,250 --> 00:06:41,260
Well that series is just a geometric
series, right?

96
00:06:41,260 --> 00:06:44,890
So this series we've already seen, in this
series converges.

97
00:06:44,890 --> 00:06:47,950
And in fact we know its value, and it's
value was 2.

98
00:06:47,950 --> 00:06:50,950
And consequently, because this condensed
series converges,

99
00:06:50,950 --> 00:06:53,900
so too must the original series converge.

100
00:06:53,900 --> 00:06:58,800
The sum of 1 over n squared as n goes from
1 to infinity converges.

101
00:06:58,800 --> 00:07:00,910
So we've seen that this series converges
by

102
00:07:00,910 --> 00:07:05,160
comparing to a telescoping series that we
know converges.

103
00:07:05,160 --> 00:07:08,210
And by using Cauchy condensation, and
there's other ways.

104
00:07:08,210 --> 00:07:10,470
We could also have used the integral test.

105
00:07:10,470 --> 00:07:14,140
So we're seeing lots of different methods
to prove the same result.

106
00:07:14,140 --> 00:07:18,690
The sum of one over n squared, n goes from
1 to infinity converges.

107
00:07:18,690 --> 00:07:20,780
And usually were happy with that, usually
were

108
00:07:20,780 --> 00:07:24,120
happy just knowing that a series converges
or diverges.

109
00:07:25,310 --> 00:07:28,840
But in this case we can ask the more
refined question.

110
00:07:28,840 --> 00:07:32,740
To what does this convergent series
converge?

111
00:07:32,740 --> 00:07:35,350
So numerically, right the sum of the first
two terms.

112
00:07:35,350 --> 00:07:40,470
1 over 1 squared plus 1 over 2 squared, is
5 4.

113
00:07:40,470 --> 00:07:41,320
And we can keep on going.

114
00:07:41,320 --> 00:07:43,760
The sum of the first 3 terms, sum of the
first

115
00:07:43,760 --> 00:07:49,440
4 terms, 5 terms, 6 terms, 7 terms, 8
terms, 9 terms.

116
00:07:49,440 --> 00:07:50,750
Little bit over 1.5.
We could

117
00:07:50,750 --> 00:07:54,230
try something a lot more terms, too,
right?

118
00:07:54,230 --> 00:07:56,215
Here's the sum of the first 100 terms,
about

119
00:07:56,215 --> 00:08:00,500
1.63, sum of the first 200 terms, first
300, 400.

120
00:08:00,500 --> 00:08:04,410
And then if we add up the first 1000
terms, right?

121
00:08:04,410 --> 00:08:06,850
So 1 over 1 plus 1 over 2 squared plus 1

122
00:08:06,850 --> 00:08:11,240
over 3 squared da, da, da plus 1 over 1000
squared.

123
00:08:11,240 --> 00:08:13,550
And we're getting just above 1.64.

124
00:08:13,550 --> 00:08:16,040
The exact

125
00:08:16,040 --> 00:08:18,260
answer is really pretty surprising.

126
00:08:18,260 --> 00:08:23,000
Numerically right we're getting about 1.64
after adding about the first 1000 terms.

127
00:08:24,250 --> 00:08:28,714
Pie squared over 6 is about 1.64 and it
turns out that

128
00:08:28,714 --> 00:08:33,806
the sum of the reciprocals of the squares
is pie squared over 6.

129
00:08:33,806 --> 00:08:40,470
This series evaluates to pie squared over
6 is a shocking result.

130
00:08:40,470 --> 00:08:41,310
It should

131
00:08:41,310 --> 00:08:44,860
really leave you wondering why, why is
something like that true?

132
00:08:44,860 --> 00:08:49,374
And unfortunately we have to wonder just a
little bit longer.

133
00:08:49,374 --> 00:08:59,374
[NOISE]