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\begin{align*} + \E(X) &= \sum_{n = 1}^{\infty} \P(X \geq n)\\ + &= \sum_{n = 1}^{\infty} \sum_{k = n}^{\infty} \mathbbm{p}^X(k) \\ + &= \sum_{n = 1}^{\infty} \sum_{k = n}^{\infty} (1-p)^{k - 1}p\\ + &= \sum_{n = 1}^{\infty} p(1-p)^{n-1}\sum_{k = 0}^{\infty} (1-p)^k + \intertext{geometrische Reihe} + &= \sum_{n = 1}^{\infty} p(1-p)^{n-1} \frac{1}{1-(1-p)}\\ + &= \sum_{n = 1}^{\infty} (1-p)^{n-1}\\ + \intertext{geometrische Reihe} + &= \frac{1}{1 - (1-p)}\\ + &= \frac{1}{p} \end{align*} \end{enumerate} \end{aufgabe}