Question Number 200931 by Spillover last updated on 26/Nov/23
$$\frac{\mathrm{1}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{18}}+\frac{\mathrm{1}}{\mathrm{30}}+\frac{\mathrm{1}}{\mathrm{45}}+\frac{\mathrm{1}}{\mathrm{63}}+\frac{\mathrm{1}}{\mathrm{84}}+...\infty=? \\ $$$$ \\ $$
Answered by MM42 last updated on 26/Nov/23
$${s}_{{n}} =\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\right)=\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}−\frac{\mathrm{1}}{{n}+\mathrm{2}}\right)\:\: \\ $$$$\Rightarrow{s}_{{n}} =\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{{n}+\mathrm{2}}\right)\:\overset{{n}\rightarrow\infty} {\Rightarrow}\:{s}=\frac{\mathrm{1}}{\mathrm{3}}\:\:\checkmark \\ $$$$ \\ $$$$ \\ $$