proof that $\eta (1)=ln2$

Theorem 1.

$\eta (1)=\ln 2$, where $\eta$ is the Dirichlet eta function.

Proof. By definition,

$$\eta (1)=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n+1}}{n}=-\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{n}.$$

Applying Abel’s Limit Theorem,

$$\eta (1)=-\underset{r\to 1^{-}}{lim}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-r)}^n}{n}=\underset{r\to 1^{-}}{lim}\ln (1+r)=\ln 2$$