let-U-n-p-k-1-n-1-n-k-p-1-with-n-p-from-N-1-calculate-lim-n-U-n-p-for-p-2-2-prove-that-U-n-1-is-convergent-3-let-V-n-k-1-n-sin-1-n-k-2-find-lim-V

Tinku Tara June 4, 2023 Relation and Functions 0 Comments

Question Number 29503 by abdo imad last updated on 09/Feb/18

let U_(n,p) = Σ_(k=1) ^n (1/((n+k)^(p+1) )) with n,p from N^★ 1) calculate lim_(n→+∞) U_(n,p) for p≥2 2)prove that U_(n,1) is convergent 3) let V_n = Σ_(k=1) ^n sin((1/((n+k)^2 ))) find lim_∞ V_n .

$${let}\:{U}_{{n},{p}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({n}+{k}\right)^{{p}+\mathrm{1}} }\:{with}\:{n},{p}\:{from}\:{N}^{\bigstar} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n},{p}} \:{for}\:{p}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{U}_{{n},\mathrm{1}} \:{is}\:{convergent} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{V}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{\mathrm{1}}{\left({n}+{k}\right)^{\mathrm{2}} }\right)\:{find}\:{lim}_{\infty} \:{V}_{{n}} \:. \\ $$

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