I am trying to use Mathematica to simplify a symbolic expression involving sum. Particularly, I define a sum via

y = (x - x0)^α Sum[a[n] (x - x0)^n, {n, 0, Infinity}]

Then I take the derivative of y with respect to x and simplify via

FullSimplify[D[y, x]]

This yields $$(x-\text{x0})^{\alpha -1} \left(\alpha \sum _{n=0}^{\infty } a(n) (x-\text{x0})^n+(x-\text{x0}) \sum _{n=0}^{\infty } n a(n) (x-\text{x0})^{n-1}\right)$$ However, the expression above can be easily simplified further to $$(x-\text{x0})^{\alpha -1} \sum _{n=0}^{\infty } ( a(n) (x-\text{x0})^n(\alpha+n) ) $$

Is there a way to "make" Mathematica recognise this simplification? I presume the problem has something to do with the fact that I use the unknown function a[n] in the expression, but I am not sure what can I do about it to get similar functionality.

I am quite new to Mathematica. Apologies if this is a trivial question.