Question Number 28617 by abdo imad last updated on 27/Jan/18
$${let}\:{give}\:{a}\:{sequence}\:{of}\:{reals}\:\left({a}_{{n}} \right)_{{n}} \:\:/\:{a}_{{n}} >\mathrm{0}\:\:{and} \\ $$ $${U}_{{n}} =\:\:\:\frac{{a}_{{n}} }{\left(\mathrm{1}+{a}_{\mathrm{1}} \right)\left(\mathrm{1}+{a}_{\mathrm{2}} \right)....\left(\mathrm{1}+{a}_{{n}} \right)} \\ $$ $$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} \:{converges} \\ $$ $$\left.\mathrm{2}\right)\:{calculate}\:\Sigma\:{u}_{{n}} \:\:{if}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\sqrt{{n}}}\:. \\ $$