I am trying to use *Mathematica* to simplify a symbolic expression involving `Sum`. The expression is defined as follows:

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

I am trying to use `FullySimplify` on the derivative of the expression with respect to `x` 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 new to *Mathematica* and would like to apologies if this is a trivial question.