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Question Number 162265 by HongKing last updated on 28/Dec/21

$$\mathrm{Calculate}:\underset{\mathbf{k}=1}{\sum ^{\mathrm{\infty }}}\frac{{\mathrm{H}}_{2\mathbf{k}}{(-1)}^{\mathbf{k}-1}}{2\mathrm{k}+1}$$$$\mathrm{where},{\mathrm{H}}_{\mathbf{n}}\mathrm{is}\mathrm{the}\mathrm{n}-\mathrm{th}\mathrm{harmonic}\mathrm{number}$$

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