Convergent power series/R/Primitive function/Fact

Let Failed to parse (syntax error): {\displaystyle {{}} f = \sum_{n <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Convergent power series/R/Primitive function/Fact]] __NOINDEX__ 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