Question Number 75985 by Ajao yinka last updated on 21/Dec/19
Answered by MJS last updated on 21/Dec/19
$$\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }=\left(\frac{{n}^{\mathrm{2}} +{n}+\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}\right)^{\mathrm{2}} \\ $$$${S}_{{k}} =\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{{n}^{\mathrm{2}} +{n}+\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}=\frac{{k}\left({k}+\mathrm{2}\right)}{{k}+\mathrm{1}} \\ $$$${S}_{\mathrm{2011}} =\frac{\mathrm{4048143}}{\mathrm{2012}} \\ $$
Commented by Ajao yinka last updated on 22/Dec/19
Excellent