Making an infinite generating function a finite one If we have some generating function $G(x)$ that generates terms indefinitely, is there a way to translate it to be a finite generating function?
For example if I only want to generate the first $k$ terms of a sequence, can I do $G(x) - x^kG(x)$ or something similar? This isn't the right answer but it's where my thought process is. Trying to find some way to "start" the recurrence at a later point so that when I subtract one infinite generating function from the other, all the terms past $k$ drop out.

Edited Jan 27 2018. Answer by M.Scheuer is sufficient.
