Convergence depends on x.
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Let's consider a power series.
So maybe I've got a power series the sum n
goes from zero to infinity of some coefficients a sub n.
Times x to the nth power.
Now suppose that I know that that
power series converges absolutely when x equals three.
So yeah, I'm going to assume that it converges absolutely at x equals
three, and then I want to know what about our other points.
What about at, say, x equals two? Does it converge there?
Well then it must converge when x equals two.
Let's see why.
Well since it converges absolutely at x equals three, that just means that the sum
n from zero to infinity of the absolute value of a sub n times three to the n.
Right?
This series converges. We can compare
this to the same series when x equals two.
But what I mean to say is just that zero is
less than or equal to the absolute value of a sub n
times two to the n, which is less than or equal to
the absolute value of a sub n times three to the n.
And since this series, the sum of a sub n times
three to the n converges, that means by the comparison test,
this series, the sum n goes fromzero0 to infinity of a sub n times two
to the n, this series converges. Which is just to say
that the original series when x equals two?
Well, in that case, this series converges absolutely.
And of course, there's nothing special about the number two.
So if x is any value so that the absolute value of x is less than or equal to three.
That just means that x is in the interval from minus three to three.
If x is any value in the interval then zero is less or equal to the absolute
value of ace of n times x to the n. Well that's just because it's the absolute
[UNKNOWN].
But then that is less than or equal to the absolute value of a sub n Times 3 to
the N so again by comparison I, that means that
this series the sum N goes from 0 to infinity.
I'll just write A sub X to the N converges absolutely.
Because the sum of the absolute value's converge
is because I'm comparing with this convergence series.
And this is the usual case. This is usually what happens.
So, talk about this though.
Let me be a little bit more formal. Let me give a name to this.
Let's call C the collection of all
real numbers so that this power series converges.
Or in words, C is all the real numbers x'd, so that this series converges.
It's a collection of numbers. The big deal here is that
C is an interval.
Well here's the theorem, this collection of values of x, where the
power series converges It turns out that collection of points is an interval,
by which I mean maybe if this open interval, maybe it's a
closed interval or maybe it's something
more complicated like some half open interval.
We'll see a proof of that soon.
And since it's an interval, this collection
of points where the power series converges is called the interval of convergence.
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