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Question Number 34596 by abdo mathsup 649 cc last updated on 08/May/18

1) prove that  Σ_(k=1) ^n  H_k =(n+1)H_n  −n  2) prove that  Σ_(k=1) ^n  H_k ^2   =(n+1)H_n ^2   −(3n+1)H_n  +2n  H_n =Σ_(k=1) ^n   (1/k) .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} =\left({n}+\mathrm{1}\right){H}_{{n}} \:−{n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:\:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:\:−\left(\mathrm{3}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}}\:. \\ $$

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