Generating functions question

Question

How to transform $\sum_{n=2}^\infty n^2x^n$ to $\sum_{n=1}^\infty nx^n$

Answer 1

$f(x)= \sum\limits_{n=2}^{\infty} n^{2}x^{n}$ and $g(x)= \sum\limits_{n=1}^{\infty} nx^{n}$ then $g(x)=nx+ \int_0^x \frac {f(t)} t dt$.