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Question Number 27666 by abdo imad last updated on 12/Jan/18

let give I_n = ∫_0 ^1   (x^n /(1+x^n ))dx  (1) prove that  lim_(n−>∝) I_n =0  (2)calculate I_n  +I_(n+1)   (3) find  Σ_(n=1) ^∝ (((−1)^(n−1) )/n) .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }{dx} \\ $$ $$\left(\mathrm{1}\right)\:{prove}\:{that}\:\:{lim}_{{n}−>\propto} {I}_{{n}} =\mathrm{0} \\ $$ $$\left(\mathrm{2}\right){calculate}\:{I}_{{n}} \:+{I}_{{n}+\mathrm{1}} \\ $$ $$\left(\mathrm{3}\right)\:{find}\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}\:. \\ $$

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