Finding the radius.
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Lets
suppose that I've been given a power series.
Perhaps is the power series, the sum and goes from one
to infinity of x to the n divided by n squared.
How do I find the radius of convergence?
Well lets try the ratio test.
So I'm going to look for absolute convergence.
So I'm really trying to figure out for which values
of x that series converges.
And I'm going to look at the ratio of the n plus first term over the n term.
So the n plus first term.
It is x to the n plus 1 over n plus 1 squared.
And I'm going to divide that by the n term which is exactly what I've got there.
X to the n over n squared. Now I can simplify that fraction a bit.
So this is. The limit n goes to infinity of, I've
got x to the n plus 1 over x to the n. So I'll just write absolute value of x.
And then I've got n plus 1 squared but it's in the denominator of the numerator.
And I've got n squared in the denominator of the denominator.
So I can write this as n squared over N plus 1 squared.
Now, what is this limit?
Well when n is very large, this quantity here is very close to one.
And this x doesn't depend on n at all.
So this limit is just the absolute value of x.
And this is the ratio between the n plus first and the nth term.
So to get absolute conversions of this series,
it's enough for ratio test that this be
less than one.
But I also know something about when the series diverges.
By the ratio test, when this limit which is the absolute value of x.
When that limit is bigger than one, then this series diverges.
Supporting it altogether what's the radius of convergence.
So to think about that let's draw a diagram.
You've got a number line. And what I know is that when the
absolute value of x is less than 1 then the series converges absolutely.
So that tells me that the series converges when x is between minus 1 and 1.
I also know that when the absolute value of
x is bigger than 1 that the series diverges.
That tells me the series diverges when x is bigger than
1 and the series diverges when x is less than minus 1.
So it converges in between here, it diverges
to right of this and diverges to the left of this.
Admittedly I haven't thought about what happens at
x equals minus 1 and x equals 1.
But I don't need to if all care about is knowing the radius of convergence.
Alright, I'm thinking about this interval being where the power series converges
and maybe it convergences at minus 1, maybe it converges at 1.
But what's the radius of this interval or
was this an interval centered at 0. And its radius is 1.
And that tells me that the
radius of conversion is 1.
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