Question Number 144484 by phally last updated on 25/Jun/21
Answered by Olaf_Thorendsen last updated on 25/Jun/21
$$\mathrm{S}_{{n}} \:=\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}\left({k}+\mathrm{2}\right)}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{S}_{{n}} \:=\:\underset{{k}=\mathrm{2}} {\overset{{n}+\mathrm{1}} {\sum}}\frac{\left({k}−\mathrm{1}\right)\left({k}+\mathrm{1}\right)}{{k}^{\mathrm{2}} } \\ $$$$\mathrm{S}_{{n}} \:=\:\underset{{k}=\mathrm{2}} {\overset{{n}+\mathrm{1}} {\sum}}\left(\mathrm{1}−\frac{\mathrm{1}}{{k}^{\mathrm{2}} }\right) \\ $$$$\mathrm{S}_{{n}} \:=\:{n}+\mathrm{1}−{H}_{{n}+\mathrm{1},\mathrm{2}} \\ $$
Commented by mnjuly1970 last updated on 26/Jun/21
$$\:{divergent}… \\ $$