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Convergence depends on x.

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Let's consider a power series.

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So maybe I've got a power series the sum n

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goes from zero to infinity of some
coefficients a sub n.

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Times x to the nth power.

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Now suppose that I know that that

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power series converges absolutely when x
equals three.

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So yeah, I'm going to assume that it
converges absolutely at x equals

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three, and then I want to know what about
our other points.

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What about at, say, x equals two?
Does it converge there?

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Well then it must converge when x equals
two.

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Let's see why.

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Well since it converges absolutely at x
equals three, that just means that the sum

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n from zero to infinity of the absolute
value of a sub n times three to the n.

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Right?

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This series converges.
We can compare

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this to the same series when x equals two.

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But what I mean to say is just that zero
is

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less than or equal to the absolute value
of a sub n

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times two to the n, which is less than or
equal to

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the absolute value of a sub n times three
to the n.

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And since this series, the sum of a sub n
times

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three to the n converges, that means by
the comparison test,

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this series, the sum n goes fromzero0 to
infinity of a sub n times two

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to the n, this series converges.
Which is just to say

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that the original series when x equals
two?

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Well, in that case, this series converges
absolutely.

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And of course, there's nothing special
about the number two.

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So if x is any value so that the absolute
value of x is less than or equal to three.

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That just means that x is in the interval
from minus three to three.

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If x is any value in the interval then
zero is less or equal to the absolute

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value of ace of n times x to the n.
Well that's just because it's the absolute

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[UNKNOWN].

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But then that is less than or equal to the
absolute value of a sub n Times 3 to

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the N so again by comparison I, that means
that

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this series the sum N goes from 0 to
infinity.

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I'll just write A sub X to the N converges
absolutely.

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Because the sum of the absolute value's
converge

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is because I'm comparing with this
convergence series.

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And this is the usual case.
This is usually what happens.

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So, talk about this though.

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Let me be a little bit more formal.
Let me give a name to this.

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Let's call C the collection of all

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real numbers so that this power series
converges.

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Or in words, C is all the real numbers
x'd, so that this series converges.

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It's a collection of numbers.
The big deal here is that

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C is an interval.

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Well here's the theorem, this collection
of values of x, where the

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power series converges It turns out that
collection of points is an interval,

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by which I mean maybe if this open
interval, maybe it's a

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closed interval or maybe it's something

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more complicated like some half open
interval.

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We'll see a proof of that soon.

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And since it's an interval, this
collection

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of points where the power series converges
is called the interval of convergence.

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