1
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Let's sum a series.
I

2
00:00:08,220 --> 00:00:10,780
want to evaluate this series carefully.

3
00:00:10,780 --> 00:00:13,820
So the claim here is that the sum k goes
from 0

4
00:00:13,820 --> 00:00:17,090
to infinity of 1 over 2 to the k is equal
to 2.

5
00:00:17,090 --> 00:00:20,550
But what does that even mean?
Well, what it really means is this.

6
00:00:21,810 --> 00:00:24,570
It means that the sum k goes from 0 to n
of one

7
00:00:24,570 --> 00:00:28,570
over to two to the k is close to 2
whenever n is big.

8
00:00:28,570 --> 00:00:28,810
Alright.

9
00:00:28,810 --> 00:00:31,220
This sum makes sense because this is just
a finite

10
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sum it's just telling me to add one over
two

11
00:00:33,220 --> 00:00:36,240
to the k for the values for the values of
k between 0 and n.

12
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So this at least makes sense.

13
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And what is close to two mean?

14
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Well, it means as close as you want it to
be to 2.

15
00:00:42,020 --> 00:00:44,410
And how big does n have to be?

16
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Well, it has to be true whenever n is
larger than some fixed big

17
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value that depends on how close you want
the thing to get to 2.

18
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Let's do the same thing, but in terms of
limits.

19
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So more precisely,

20
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I'll define the sequence, s sub n.
The sequence of partial sums.

21
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To be the sum, k goes from 0 to n of 1
over 2 to the k.

22
00:01:09,950 --> 00:01:13,730
And the claim that this series has value
2.

23
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Just means that this sequence has limit 2.
It means that the limit

24
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as n approaches infinity of the nth
partial sum is 2.

25
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How do I evaluate that limit?
Well, let's look for a pattern.

26
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So s sub 0 lets the sum k goes from 0 to 0
of 1 over 2 of the k.

27
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So that's just 1 over 2 to the 0, which is
just 1.

28
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S sub 1, well that's the sum k goes from 0
to 1 of 1 over 2 of the k.

29
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That's 1 plus 1 over 2 to the first power,
which is a half.

30
00:01:49,060 --> 00:01:49,560
S sub

31
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2, well that's the sum, k goes from 0 to
2, of 1 over 2 to the k.

32
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That's 1 plus a half, plus 1 over 2
squared, which is a fourth.

33
00:01:59,600 --> 00:02:00,570
I could simplify that a little bit.

34
00:02:00,570 --> 00:02:05,580
I could write that as one plus 3 4th s sub
3.

35
00:02:05,580 --> 00:02:08,680
And that's the sum k goes from 0 to 3 of 1
over 2 to

36
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the k, so that's 1 plus a half plus a 4th
plus 1 over 2

37
00:02:14,810 --> 00:02:18,110
to the 3rd, which is an 8th.

38
00:02:18,110 --> 00:02:21,005
And I could add a half, a 4th, and 8th and
get 7 8th.

39
00:02:22,160 --> 00:02:24,535
Then I could compute s sub 4, right?

40
00:02:24,535 --> 00:02:27,015
And that would be 1 plus a half plus a
fourth plus

41
00:02:27,015 --> 00:02:30,253
an 8th plus 1 16th, and that'll be 1 plus
15 16th.

42
00:02:30,253 --> 00:02:34,168
And now maybe you're beginning to the
beginnings of the pattern here, right?

43
00:02:34,168 --> 00:02:35,890
I've got a half, 3 quarter, 7 8th, 15 16th
here.

44
00:02:35,890 --> 00:02:40,620
So it's perhaps believable.
I mean, this isn't a proof,

45
00:02:40,620 --> 00:02:45,616
it's just a little bit of evidence but the
pattern is suggesting itself.

46
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That the n-th partial sum is 1 plus,
let's, suppose

47
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to be a number a little bit less than one.

48
00:02:51,250 --> 00:02:54,190
And it's the corresponding power of two in

49
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the denominator, and one less in the
numerator.

50
00:02:58,050 --> 00:03:05,660
So I could write that as 1 plus 2 to the n
minus one over 2 to the n.

51
00:03:05,660 --> 00:03:05,760
Now

52
00:03:05,760 --> 00:03:07,570
I can simplify that a little bit.

53
00:03:07,570 --> 00:03:15,080
Let's split that up as 1 plus 2 to the n
over 2 to the n minus 1 over 2 to the n.

54
00:03:15,080 --> 00:03:19,770
And I can write that as 2 minus 1 over 2
to the n.

55
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I haven't proved that, but hopefully that
formula seems believable.

56
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In any case, armed with that formula, we
can evaluate the limit.

57
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Well, here we go.

58
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I want to take the limit

59
00:03:31,840 --> 00:03:35,710
of S sub n as n approaches infinity, and

60
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now I've got that formula for the nth
partial sum.

61
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So that's the limit as n approaches
infinity of 2 minus 1 over 2 to the n.

62
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That's a limit of a difference, which is

63
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the difference of a limit, provided limits
exist.

64
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So this is the limit just of 2 as it

65
00:03:52,510 --> 00:03:56,930
approaches infinity minus the limit of 1
over 2 to the

66
00:03:56,930 --> 00:03:59,630
n as n approaches infinity.

67
00:03:59,630 --> 00:04:04,080
This is the limit of a constant, which is
just that constant minus,

68
00:04:04,080 --> 00:04:06,910
this is the limit of a quotient, but look
at what happens here.

69
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The denominator is very, very large.

70
00:04:09,190 --> 00:04:11,800
I can make the denominator as large as I
like.

71
00:04:11,800 --> 00:04:14,290
Right?
2 to the n can be very, very positive.

72
00:04:14,290 --> 00:04:19,370
1 over a very large number is very close
to 0.

73
00:04:19,370 --> 00:04:22,150
So this limit is, in fact, zero.

74
00:04:22,150 --> 00:04:25,890
And that means that the limit of the
partial sums is just two.

75
00:04:25,890 --> 00:04:28,930
That means that this series converges to
two.

76
00:04:28,930 --> 00:04:34,170
And I can not only see that algebraically
by, by using that limit.

77
00:04:34,170 --> 00:04:36,500
But I can also see it geometrically.

78
00:04:36,500 --> 00:04:38,950
Geometrically, I can draw a picture like
this.

79
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Where I got a half.

80
00:04:40,840 --> 00:04:43,740
A quarter of the square, an eighth of the
square, a 16th

81
00:04:43,740 --> 00:04:47,320
of the square, a 32nd of the square, a
64th of the square.

82
00:04:47,320 --> 00:04:50,940
And all these pieces fit together, to
build one unit square.

83
00:04:50,940 --> 00:04:54,740
And what that's showing me, is that the
sum, k goes from 1

84
00:04:54,740 --> 00:04:58,570
to infinity, of 1 over 2 to the k, well,
that really is 1.

85
00:04:58,570 --> 00:05:03,840
So the sum, k goes from 0 to infinity, of
1 over 2 to the k Well,

86
00:05:03,840 --> 00:05:08,220
that's the first term plus all the rest of
these terms, so that's 1 plus 1.

87
00:05:08,220 --> 00:05:13,772
That must

88
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be 2

89
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[SOUND]