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advanced-calculus-evaluate-n-1-H-n-2-n-2-where-H-n-k-1-n-1-k-harmonic-number-




Question Number 121774 by mnjuly1970 last updated on 11/Nov/20
          ... advanced  calculus...         evaluate:                            Σ_(n=1) ^∞ (((H_n )^2 )/n^2 ) =?      where   H_n =Σ_(k=1) ^n (1/k) (harmonic number)
$$\:\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:{evaluate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left({H}_{{n}} \right)^{\mathrm{2}} }{{n}^{\mathrm{2}} }\:=? \\ $$$$\:\:\:\:{where}\:\:\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\:\left({harmonic}\:{number}\right) \\ $$

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