Dirichlet eta function

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

$$\eta (s):=\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n-1}}{n^s}.$$(1)

Let $s=\sigma +it$. For $s$ a positive real number the series converges by the alternating series test, by the second 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 zeroof $1-2^{1-s}$.