Integrating a series and finding its sum

Given:

Radius of convergence is 2.

Defined as a function:

Show that for the following is true:

My attempt:

I first start by changing the definition under the sum from to :

To integrate a series, I use the following formula:

Using that formula I get:

But now I have no idea what to do next.

I know that the sum for a power series is:

But the problem with this formula is that it only counts for whereas I have

I also don't know how to express my series as .

Re: Integrating a series and finding its sum

Note that

This geometric serie converges if with sum

therefore (with the factor ) the sum is