1
00:00:00,300 --> 00:00:03,990
Infinite radius.

2
00:00:08,220 --> 00:00:12,660
It can certainly happen that the radius of
convergence is infinite.

3
00:00:12,660 --> 00:00:14,710
Well, let's consider this power series.

4
00:00:14,710 --> 00:00:20,180
The sum, n goes from 0 to infinity, of x
to the n

5
00:00:20,180 --> 00:00:25,960
divided by n factorial.
Let's try the ratio test.

6
00:00:25,960 --> 00:00:27,220
So, here we go.

7
00:00:27,220 --> 00:00:31,260
the limit n goes to infinity of the n plus
first term.

8
00:00:31,260 --> 00:00:33,700
So, x to the n plus 1

9
00:00:33,700 --> 00:00:36,330
over n plus 1 factorial divided by just

10
00:00:36,330 --> 00:00:38,930
the nth term which is what's displayed
here.

11
00:00:38,930 --> 00:00:42,890
X to the n over n factorial and absolute
value of

12
00:00:42,890 --> 00:00:46,180
it 'because I'm checking for absolute
convergence of the ratio test.

13
00:00:46,180 --> 00:00:47,320
I can simplify this.

14
00:00:47,320 --> 00:00:49,020
If it is a fraction with fractions in the

15
00:00:49,020 --> 00:00:52,920
numerator and denominator, this is the
limit n goes

16
00:00:52,920 --> 00:00:55,670
to infinity of x to the n plus 1

17
00:00:55,670 --> 00:00:58,960
times n factorial in the denominator of
the denominator,

18
00:00:58,960 --> 00:01:00,580
put that in the numerator.

19
00:01:00,580 --> 00:01:06,800
Divided by x to the n times n plus 1
factorial.

20
00:01:06,800 --> 00:01:09,030
So, i've just got a fraction, but I can
simplify this

21
00:01:09,030 --> 00:01:13,340
a bit too, this is the limit n goes to
infinity.

22
00:01:13,340 --> 00:01:16,320
We've got x to the n plus 1 over x to the
n.

23
00:01:16,320 --> 00:01:19,030
Which just leaves me with an x in the
numerator, and I've got

24
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n factor on the numerator, and an n plus 1
factorial in the denominator.

25
00:01:23,320 --> 00:01:24,420
Well, n plus 1

26
00:01:24,420 --> 00:01:26,995
factorial kills everything here, right?

27
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N plus 1 factorial is 1 times 2, all the
way through

28
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n plus 1, and that contains all the terms
in n factorial.

29
00:01:34,410 --> 00:01:38,080
So, what I'm left with, is just an n plus
1 and the denominator.

30
00:01:38,080 --> 00:01:39,190
Now, what is this limit?

31
00:01:39,190 --> 00:01:42,470
X is just some fixed quantity, it doesn't
depend on n.

32
00:01:42,470 --> 00:01:44,420
But what's the limit then, of some number
x

33
00:01:44,420 --> 00:01:47,890
divided by n plus 1, n going to infinity?

34
00:01:47,890 --> 00:01:49,370
Well, this limit is zero.
Alright?

35
00:01:49,370 --> 00:01:51,410
It doesn't matter what x is.

36
00:01:51,410 --> 00:01:53,870
If you take some fixed number and divide
it by a very

37
00:01:53,870 --> 00:01:57,410
large quantity, you can make this as close
to zero as you'd like.

38
00:01:57,410 --> 00:02:01,140
So, this limit is zero.
Zero is less than one.

39
00:02:01,140 --> 00:02:08,250
And that means by the ratio test, this
series converges regardless of what x is.

40
00:02:10,200 --> 00:02:14,700
So, what's the radius of convergence?
Series converges for

41
00:02:14,700 --> 00:02:16,300
all values of x.

42
00:02:16,300 --> 00:02:20,724
And that means the radius of convergence
is infinity.

43
00:02:20,724 --> 00:02:26,680
And in the not to distant future, we're
going to see a very surprising result.

44
00:02:26,680 --> 00:02:30,310
We're eventually going to see that this
series.

45
00:02:30,310 --> 00:02:35,160
Is in fact a complicated way of writing
down a function we already know.

46
00:02:35,160 --> 00:02:39,980
This is just a complicated way of writing
down the function e to the x.

47
00:02:39,980 --> 00:02:42,830
Meaning that if you plug in specific
values

48
00:02:42,830 --> 00:02:46,570
for x, say x equals negative 1, you'd get

49
00:02:46,570 --> 00:02:53,110
that the sum n goes from 0 to infinity of
minus 1 to the n over n factorial.

50
00:02:53,110 --> 00:02:57,340
Is equal to e to the negative 1, it's
equal to 1 over e.

51
00:02:57,340 --> 00:03:00,620
And by choosing different values of x, by,
by plugging in different

52
00:03:00,620 --> 00:03:05,600
specific values of x, we can generate a
ton of really neat series.

53
00:03:05,600 --> 00:03:06,730
Like, here's another example.

54
00:03:06,730 --> 00:03:10,030
I mean, just the fact that the sum, n goes
from 0 to

55
00:03:10,030 --> 00:03:15,920
infinity of 10 to the n over n factorial,
is, well, according to this.

56
00:03:15,920 --> 00:03:19,590
That's just e to the 10th power.
And, that is really cool.

57
00:03:19,590 --> 00:03:21,150
But, let me warn you.

58
00:03:21,150 --> 00:03:24,680
And, share a bit of the philosophy of the
power series with you.

59
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Yes, by plugging in specific values of x,

60
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you can generate a ton of interesting
examples.

61
00:03:30,540 --> 00:03:32,700
But, power series aren't just a way

62
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of generating a bunch of series and
isolations.

63
00:03:35,620 --> 00:03:39,460
Part of the joy of power series comes by
thinking of power

64
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series, not as a, a mechanism for
generating a bunch of discrete examples.

65
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But as a way of collecting together a
whole

66
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bunch of interesting series that depend on
a parameter x.

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[SOUND]

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[SOUND]