As the title says, I'm having a lot of trouble finding a power series representation of a function. Here it is:

And here's my attempt at a solution:

I had a lot of trouble trying to reindex the summations such that I'd have a single summation at the end. This is what I ultimately got. Is this correct? If so, how would I go about finding the interval of convergence here? Would it just be with the ratio test as usual? It seems like it would be quite difficult to do that here, considering the number of terms in the final summation.