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Question Number 34721 by abdo mathsup 649 cc last updated on 10/May/18

let ξ(x) = Σ_(n=1) ^∞  (1/n^x )  with x>1  1)prove that  (1/(x−1)) ≤ξ(x)≤ (x/(x−1))  2) find lim_(x→1^+ ) (x−1)ξ(x) .

$${let}\:\xi\left({x}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:{with}\:{x}>\mathrm{1} \\ $$ $$\left.\mathrm{1}\right){prove}\:{that}\:\:\frac{\mathrm{1}}{{x}−\mathrm{1}}\:\leqslant\xi\left({x}\right)\leqslant\:\frac{{x}}{{x}−\mathrm{1}} \\ $$ $$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \left({x}−\mathrm{1}\right)\xi\left({x}\right)\:. \\ $$

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