For $s\in\mathbb{C}$ , the Dirichlet eta function is defined as

\begin{equation} \eta(s) := \sum_{n=1}^{\infty} \frac{\left( -1 \right)^{n-1}}{n^{s}}\,. \end{equation} Let $s=\sigma + it$ . For $s$ a positive real number the series converges by the alternating series test, by the second property listed in the entry on Dirichlet series it converges for all $s$ with $\sigma > 0$ .

It can be shown that $\eta(s) = (1-2^{1-s})\zeta(s)$ , where $\zeta(s)$ is the Riemann zeta function. The pole of $\zeta(s)$ at $s=1$ is cancelled by the zero of $1-2^{1-s}$ .