Expectation of Geometric Distribution/Formulation 1/Proof 3

Proof
From the definition of expectation:


 * $\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$

Then

By the Ratio Test, both $\sum_{k \mathop \ge 1} k p^k$ and $\sum_{k \mathop \ge 1} k p^{k+1}$ converge absolutely.

From Absolutely Convergent Real Series is Convergent, both series converge.