Convergent power series/R/Primitive function/Fact

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

Then the power series

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

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

for ${}f$.

Write a proof