# Power series

1. Jun 20, 2013

### aaaa202

At exam today I was to calculate an improper integral of a function f defined by a power series.
Inside this radius you could of course integrate each term, i.e. symbologically:
∫Ʃ = Ʃ∫
The only problem is that the improper integral went from 0 to 1.
Is it then true that:
limx->1[∫Ʃ ]=limx->1[Ʃ∫]
and what theorem assures this? At the exam I didn't think about this unfortunately, but I would probably not have known what to do anyways. I think there is a theorem called Abels theorem which shows that a power series is continous also in x=±r, but I'm not sure if that's what im looking for.

2. Jun 20, 2013

### Zondrina

Are you familiar with uniform convergence?

Abel's theorem basically says that if a series converges at x = R, it converges uniformly over the interval
0 ≤ x ≤ R.

As a consequence, you can also observe uniform convergence over the interval -R ≤ x ≤ 0.

A corollary would be that it converges on -R ≤ x ≤ R.

Last edited: Jun 20, 2013
3. Jun 20, 2013

### aaaa202

Yes I am familiar with uniform convergence. And you agree that from what I told you I would have had to invoke Abels theorem to give a satisfactory answer?

4. Jun 20, 2013

### Zondrina

Uniform convergence would be a sufficient condition for you to be able to switch limits/derivatives/integrals around. So yes, Abel's theorem could be used here as it guarantees the uniform convergence over the whole interval [-R,R].

5. Jun 20, 2013

### Office_Shredder

Staff Emeritus
I don't think you need anything fancy at all. Assuming that your variable x is "integrate from 0 to x" then for every x<1, the thing you have written in square brackets [ ] is equal whether the integration or the summation comes first, because you're integrating over [0,x] which is bounded away from 1. You just did algebra inside the limit which is legal for the values of x that you are considering

6. Jun 20, 2013

### aaaa202

Im switching 2 limits:
limx->1limn->∞[Ʃanxn] = limn->∞limx->1[Ʃanxn]
I only know that the power series converges for lxl<1