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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