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Question Number 147670 by mnjuly1970 last updated on 22/Jul/21

Answered by Olaf_Thorendsen last updated on 22/Jul/21

By definition H_n ^((2))  = Σ_(k=1) ^n (1/k^2 )  ⇒ H_(n−1) ^((2)) +(1/n^2 ) = Σ_(k=1) ^(n−1) (1/k^2 )+(1/n^2 ) = H_n ^((2))   Trivial.

$$\mathrm{By}\:\mathrm{definition}\:\mathrm{H}_{{n}} ^{\left(\mathrm{2}\right)} \:=\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$$$\Rightarrow\:\mathrm{H}_{{n}−\mathrm{1}} ^{\left(\mathrm{2}\right)} +\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:=\:\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} }+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:=\:\mathrm{H}_{{n}} ^{\left(\mathrm{2}\right)} \\ $$$$\mathrm{Trivial}. \\ $$

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