Calculate-k-1-H-2k-1-k-1-2k-1-where-H-n-is-the-n-th-harmonic-number-

Tinku Tara June 4, 2023 Algebra 0 Comments

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

Calculate: Σ_(k=1) ^∞ ((H_(2k) (-1)^(k-1) )/(2k + 1)) where, H_n is the n-th harmonic number

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

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Calculate-k-1-H-2k-1-k-1-2k-1-where-H-n-is-the-n-th-harmonic-number-

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