Question Number 77556 by behi83417@gmail.com last updated on 07/Jan/20
$$\mathrm{1}.\:\:\:\mathrm{1}×\mathrm{1}!×\mathrm{2}!+\mathrm{2}×\mathrm{2}!×\mathrm{3}!+\mathrm{3}×\mathrm{3}!×\mathrm{4}!+....=? \\ $$$$\mathrm{2}.\:\:\:\:\frac{\mathrm{1}+\sqrt{\mathrm{2}}}{\mathrm{2}+\sqrt{\mathrm{3}}}+\frac{\mathrm{2}+\sqrt{\mathrm{3}}}{\mathrm{3}+\sqrt{\mathrm{4}}}+\frac{\mathrm{3}+\sqrt{\mathrm{4}}}{\mathrm{4}+\sqrt{\mathrm{5}}}+......=? \\ $$$$\mathrm{3}.\:\:\:\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}}+\frac{\mathrm{2}}{\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}+\sqrt{\mathrm{4}}}+\frac{\mathrm{3}}{\sqrt{\mathrm{3}}+\sqrt{\mathrm{4}}+\sqrt{\mathrm{5}}}+....=?\: \\ $$
Commented by mr W last updated on 07/Jan/20
$$\mathrm{1}.\:\:\rightarrow\infty,\:{clear} \\ $$$$\mathrm{2}.\:{a}_{{n}} \rightarrow\frac{{n}}{{n}}\rightarrow\mathrm{1},\:\Sigma\:\rightarrow\infty \\ $$$$\mathrm{3}.\:\:{a}_{{n}} \rightarrow\frac{\sqrt{{n}}}{\mathrm{3}}\rightarrow\infty,\:\Sigma\rightarrow\infty \\ $$
Commented by behi83417@gmail.com last updated on 08/Jan/20
$$\mathrm{thak}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{dear}\:\mathrm{master}. \\ $$