Question Number 162265 by HongKing last updated on 28/Dec/21
$$\mathrm{Calculate}:\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{H}_{\mathrm{2}\boldsymbol{\mathrm{k}}} \:\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{k}}-\mathrm{1}} }{\mathrm{2k}\:+\:\mathrm{1}} \\ $$$$\mathrm{where},\:\mathrm{H}_{\boldsymbol{\mathrm{n}}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{n}-\mathrm{th}\:\mathrm{harmonic}\:\mathrm{number} \\ $$