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Question Number 33892 by math khazana by abdo last updated on 26/Apr/18

prove that  Σ_(n=1) ^∞   (H_n /n^2 ) =2 ξ(3) with  ξ(x) =Σ_(n=1) ^∞  (1/n^x )     and x>1.

$${prove}\:{that}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:=\mathrm{2}\:\xi\left(\mathrm{3}\right)\:{with} \\ $$ $$\xi\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:\:\:{and}\:{x}>\mathrm{1}. \\ $$

Commented bymath khazana by abdo last updated on 26/Apr/18

H_n = Σ_(k=1) ^n  (1/k) .

$${H}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:. \\ $$

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