Convergent power series/R/Primitive function/Fact

Let $f=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$ denote a power series which converges on $]-r,r[$.

Then the power series

$\sum _{n=1}^{\mathrm{\infty }}\frac{a_{n-1}}{n}{x}^n$

converges also on $]-r,r[$, and represents a primitive function

for $f$.

Write a proof