The series

(1)

Proof. Let $s = \sigma+it$ where $\sigma,\,t \in \mathbb{R}$ and $\sigma > 1$ . Then $$\left|\frac{1}{n^s}\right| = \frac{1}{|e^{s\log{n}}|} = \frac{1}{e^{\sigma\log{n}}} = \frac{1}{n^\sigma}.$$ Since the series $\sum_{n=1}^\infty\frac{1}{n^\sigma}$ converges, by the $p$ -test, for $\sigma > 1$ , we conclude that the series (1) is absolutely convergent in the half-plane $\sigma > 1$ .