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Radius and center.

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00:00:02,662 --> 00:00:08,810
[MUSIC]

3
00:00:08,810 --> 00:00:11,530
Well thus far, we've been considering

4
00:00:11,530 --> 00:00:15,200
power series that are centered around
zero.

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00:00:15,200 --> 00:00:18,320
By which I mean, we've been looking at
power series look like this.

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00:00:18,320 --> 00:00:22,180
The sum, n goes from zero, say to
infinity.

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00:00:22,180 --> 00:00:26,160
Of C sub n times just X to the n.

8
00:00:26,160 --> 00:00:28,030
But we can center our power series around

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00:00:28,030 --> 00:00:30,740
some other point around some other point
a.

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00:00:30,740 --> 00:00:32,700
So instead of writing this.

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00:00:32,700 --> 00:00:33,820
Let's say I want to

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00:00:33,820 --> 00:00:38,980
write down a power series that's centered
around

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00:00:38,980 --> 00:00:43,500
some other point say a.
Well then I'd write my power series like

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00:00:43,500 --> 00:00:47,944
this.
The sum n goes from 0 to infinity of C sub

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00:00:47,944 --> 00:00:53,480
n again, but now times, not just x to the
n, but x

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00:00:53,480 --> 00:00:58,990
minus a to the nth power.
In this case, the interval of convergence

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00:00:58,990 --> 00:01:01,500
is centered around a as well.

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00:01:01,500 --> 00:01:05,160
Well, let's find an interval on which this
series converges.

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00:01:05,160 --> 00:01:09,380
And the goal is to show that interval is
really centered around a so

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00:01:09,380 --> 00:01:13,460
it legitimately makes sense to think of
this power series as centered around a.

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00:01:13,460 --> 00:01:15,510
And to find the interval on which this
power

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00:01:15,510 --> 00:01:19,540
series converges I am going to use the
ratio test.

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00:01:19,540 --> 00:01:22,760
So let's, let's do that now.
So how does the ratio test work here?

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00:01:22,760 --> 00:01:24,220
Well, it asks me to look at the

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00:01:24,220 --> 00:01:29,220
limit as n approaches infinity, of the n
plus first term, c sub n plus

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00:01:29,220 --> 00:01:34,680
1 times x minus a to the n plus first
power, divided by the nth term.

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00:01:34,680 --> 00:01:37,860
Which is c sub n times x minus a to the n.

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00:01:37,860 --> 00:01:41,420
And, I've got absolute value bars here
because I'm really

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00:01:41,420 --> 00:01:44,000
applying the ratio test to the sum of the
absolute values.

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00:01:44,000 --> 00:01:46,980
I'm using it to analyze absolute
convergence.

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00:01:46,980 --> 00:01:47,920
Okay, well, what's this limit?

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00:01:47,920 --> 00:01:49,390
And well, I can simplify this limit

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00:01:49,390 --> 00:01:50,480
a bit.

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00:01:50,480 --> 00:01:54,160
I could write this as the limit, n goes to
infinity, let me

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00:01:54,160 --> 00:01:56,745
pull off the C sub n plus 1 and the C sub
n.

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00:01:56,745 --> 00:02:01,300
That'll be the fraction C sub n plus 1,
over C sub n.

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00:02:01,300 --> 00:02:04,700
And in here I've got n plus 1 copies of x
minus a.

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00:02:04,700 --> 00:02:09,080
Here I've got n copies of x minus a.
Most of those will cancel.

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00:02:09,080 --> 00:02:11,260
All that I'm left with is just one copy of
x minus

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a in the numerator, so I'll write times
absolute value of x

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00:02:14,470 --> 00:02:14,820
minus a.

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00:02:14,820 --> 00:02:19,890
And the ratio test tells me that if this
is less than 1, then the

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00:02:19,890 --> 00:02:23,490
original series converges absolutely, and
if this is

44
00:02:23,490 --> 00:02:26,059
bigger than 1, then the original series
diverges.

45
00:02:28,010 --> 00:02:30,240
Well, how can I really analyze this.

46
00:02:30,240 --> 00:02:33,230
I don't really know anything about C sub n
plus 1 and C sub n.

47
00:02:33,230 --> 00:02:34,360
But let's just pretend, right?

48
00:02:34,360 --> 00:02:37,000
Let's just pretend that the limit just of
this C sub

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00:02:37,000 --> 00:02:39,612
n plus 1 over C sub n ends up being 1

50
00:02:39,612 --> 00:02:40,610
over R.

51
00:02:40,610 --> 00:02:42,680
And in this part here, the absolute value

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00:02:42,680 --> 00:02:44,920
of x minus a, that's just a constant
anyway.

53
00:02:44,920 --> 00:02:48,330
So I'll write times absolute value x minus
a.

54
00:02:48,330 --> 00:02:51,275
I'm wondering when is that less than 1.

55
00:02:51,275 --> 00:02:54,510
Well, I mean I'm just playing make believe
here.

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00:02:54,510 --> 00:02:59,120
R is some positive number, I'll multiply
both sides of this inequality by R.

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00:02:59,120 --> 00:03:02,140
So it's positive, doesn't affect the
inequality at all.

58
00:03:02,140 --> 00:03:05,140
Now I've got the absolute value x minus a

59
00:03:05,140 --> 00:03:08,760
is less than R after multiplying through
by R.

60
00:03:08,760 --> 00:03:09,550
Well, what does this mean?

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00:03:09,550 --> 00:03:14,445
This means the distance between x and a is
less than R.

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00:03:14,445 --> 00:03:21,682
Well, that really means that x is between
a minus R and a plus R.

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00:03:21,682 --> 00:03:24,605
And then, at least when x is between a
minus R

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00:03:24,605 --> 00:03:28,607
and a plus R, I know that the series
converges absolutely.

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00:03:28,607 --> 00:03:30,631
I don't know what happens

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00:03:30,631 --> 00:03:35,599
at the end points, but the big deal here
is that look, this thing

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00:03:35,599 --> 00:03:41,452
legitimately looks like an interval
centered at the point a with radius R.

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00:03:41,452 --> 00:03:44,962
In what's to come, I'll usually be talking
about power series

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00:03:44,962 --> 00:03:48,667
that are centered around zero, but it's
important to realize that

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00:03:48,667 --> 00:03:51,917
we can talk about power series That are
centered around some

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00:03:51,917 --> 00:03:56,082
other point and there are certainly cases
where that'll come in handy.

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[NOISE]