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Question Number 29503 by abdo imad last updated on 09/Feb/18
$${let}\:{U}_{{n},{p}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({n}+{k}\right)^{{p}+\mathrm{1}} }\:{with}\:{n},{p}\:{from}\:{N}^{\bigstar} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n},{p}} \:{for}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{U}_{{n},\mathrm{1}} \:{is}\:{convergent} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{V}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{\mathrm{1}}{\left({n}+{k}\right)^{\mathrm{2}} }\right)\:{find}\:{lim}_{\infty} \:{V}_{{n}} \:. \\ $$