I am having trouble solving a problem involving a series.

The problem is the following:

The problem wants me to rewrite the given expression as a sum who generic term involves

For some reason I can't get the same answer that is in the back of the book. I especially don't know what to do when I have the variable "n" with the variable "k". I've tried substituting (n+1) in the place of "n" but that still does work. That method worked in similar problems, so I feel I'm missing out on a subsequent step.

Can anyone please explain thi s to me?

Thank you!