I suggest to use amsthm. The name of the proof will automatically be translated to "Beweis" thanks to the babel package and it solves your additional problem with the number you mentioned [in your comment][1.
Thanks to barbara beeton for the hint with the qed symbol!
\documentclass[12pt,a4paper,twoside]{article}
\usepackage[ngerman]{babel}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\begin{document}
\begin{proof}
Es ist (geometrische Reihe):
\begin{align*}
\sum_{k=0}^{\infty}\Big(\frac{1}{4}\Big)^k=&\frac{1}{1-\frac{1}{4}}\\
=&\frac{4}{3}\\
\iff \sum_{k=1}^{\infty}\Big(\frac{1}{4}\Big)^k=&\frac{1}{3}
\end{align*}
Und damit ist bei nur n Viertelungen:
\begin{align*}
\sum_{k=1}^{\infty}\Big(\frac{1}{4}\Big)^k=\sum_{k=1}^{n}\Big(\frac{1}{4}\Big)^k+\sum_{k=n+1}^{\infty}\Big(\frac{1}{4}\Big)^k=&\frac{1}{3}\\
\iff \sum_{k=1}^{n}\Big(\frac{1}{4}\Big)^k=&\frac{1}{3}-\sum_{k=n+1}^{\infty}\Big(\frac{1}{4}\Big)^k\qedhere
\end{align*}
\end{proof}
\end{document}

amssymband notamsymbol– samcarter_is_at_topanswers.xyz Nov 02 '17 at 15:27\usepackage[thmmarks]{ntheorem} \theoremsymbol{\ensuremath\square} \newtheorem{Beweis}{Beweis}
\begin{document}
\begin{Beweis} gwe \end{Beweis}
\end{document}` gives me a square at the end.
– samcarter_is_at_topanswers.xyz Nov 02 '17 at 15:28\theoremsymbol{\ensuremath\square} \theoremseparator{:} \newtheorem{Beweis}{Beweis}
\begin{document}
\begin{Beweis} Sample Text \end{Beweis}
\end{document}
Gives me no square
– curies Nov 02 '17 at 15:39Plus another question. Is there a possibility to remove the number after proof? So it is not "Beweis 1" just "Beweis"?
– curies Nov 02 '17 at 16:08