1
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Radius zero.

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[SOUND]

3
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Nobody's guaranteeing us all that much
convergence.

4
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As an example, let's consider this power
series.

5
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The sum n goes from 0 to infinity.

6
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Of n factorial times x to the n.

7
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For sure, if x is equal to 0, then this
power series converges.

8
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Yeah, so what happens when x is equal to
0?

9
00:00:31,390 --> 00:00:35,862
Then this series is the sum n goes from 0

10
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to infinity, of n factorial times 0 to the
nth power.

11
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Now what is this sum?

12
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Right, does this series converge?

13
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Well, I prefer the convention that 0 to
the 0 is 1.

14
00:00:50,220 --> 00:00:53,688
And I think pretty much everyone prefers
the convention that

15
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0 factorial is 1.

16
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So the first term, the n equals 0 term of
this series is equal to 1.

17
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But what about the next term?
What about the n equals 1 term?

18
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Well, that's 1 factorial times 0 to the
first power, that zero.

19
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What about the next term?

20
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What about n equals 2?

21
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That's 2 factorial times 0 squared.
That's 0.

22
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What about n equals 3?

23
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That's a number times 0 to the third
power, that's zero.

24
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What about n equals 4?

25
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That's a number times 0 to the fourth
power.

26
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That's

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[INAUDIBLE].

28
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All the other terms in this series are 0.
So this series has the value 1.

29
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It, it converges at x equals 0.
But are there any

30
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other values of x, besides 0, for which
this series converges.

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So I know this converge at x equals 0.

32
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But lets suppose that x is some non-zero
number.

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Then does this series converge or diverge.

34
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Well, let's check it with the ratio test.

35
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So, I'm going to look at the limit as n
goes to infinity.

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Once the n plus first term here, that's n
plus 1 factorial times x to

37
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the n plus 1 divided by just the nth term,
which is n factorial times x to the n.

38
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Look at the absolute value of that.

39
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What's this limit?
Well I can simplify this limit, right?

40
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This is the limit of what's n plus 1
factorial over n factorial?

41
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Well I can simplify a bit.
That's just n plus 1.

42
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And then I've got x to the n plus 1 over x
to the n, that's just x.

43
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So for some fixed value of x which isn't
0.

44
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What is this limit?

45
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Well if x is anything but 0, this limit is
enormous number, times non-zero number.

46
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This is very, very positive, right?
This limit is infinite.

47
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And that is bigger than 1.

48
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What that means is that by the ratio test,
this series diverges.

49
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For any fixed value of x not 0, this
series diverges.

50
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So it doesn't converge anywhere else, it
only converges when x equals 0.

51
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So in other words, since this series only
converges at the

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point x equals 0, and diverges whenever x
is not zero.

53
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That means the radius of convergence, is
equal to 0.

54
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Let me leave you with a question.

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Try to think of other power series with
radius of convergence equal to 0.

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Can you think of any other examples?

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[SOUND]

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[SOUND]