Calculating Improper Integral w/ Power Series of r=1

[aaaa202]

At exam today I was to calculate an improper integral of a function f defined by a power series.
The power series had radius of convergence r=1.
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:
lim_x->1[∫Ʃ ]=lim_x->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 continuous also in x=±r, but I'm not sure if that's what I am looking for.

 

[STEMucator]

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.

 

[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?

 

[STEMucator]

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?

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].

 

[Office_Shredder]

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

 

[aaaa202]

Im switching 2 limits:
lim_x->1lim_n->∞[Ʃa_nxⁿ] = lim_n->∞lim_x->1[Ʃa_nxⁿ]
I only know that the power series converges for lxl<1