What is the value of alternating series which I mention below

Question

Plz tell me about this more , Is its value in terms of π ? If so then plz tell me what is that value.

Answer 1

The $\ell >0$ terms famously sum to $\ln 2$ (if you want to learn why, look up the Taylor series of $\ln (1+x)$). Since $\ell\to-\ell$ multiplies $(-1)^{\ell+1}/\ell$ by $-1$, the sum over all $\ell\in\mathbb{Z}\backslash\{0\}$ is $0$.