Question Number 3027 by Rasheed Soomro last updated on 03/Dec/15
$${Determine} \\ $$$$\mathrm{1}+\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{3}} +... \\ $$
Commented by Filup last updated on 03/Dec/15
$$=\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{i}^{−{i}} \\ $$
Commented by prakash jain last updated on 04/Dec/15
$${a}_{{n}} =\frac{\mathrm{1}}{{n}^{{n}} }\leqslant\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\mathrm{for}\:{n}\geqslant\mathrm{2} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\mathrm{converges}\:\mathrm{so}\:\mathrm{the}\:\mathrm{given}\:\mathrm{series}\:\mathrm{converges}. \\ $$