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let-S-n-k-0-1-k-2k-1-1-prove-that-pi-4-S-n-1-n-1-0-1-t-2n-2-1-t-2-dt-2-conclude-lim-n-S-n-




Question Number 48506 by Abdo msup. last updated on 24/Nov/18
let  S_n =Σ_(k=0) ^∞  (((−1)^k )/(2k+1))  1)prove that (π/4) −S_n =(−1)^(n+1)  ∫_0 ^1  (t^(2n+2) /(1+t^2 ))dt  2) conclude lim_(n→+∞)   S_n .
$${let}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\frac{\pi}{\mathrm{4}}\:−{S}_{{n}} =\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{t}^{\mathrm{2}{n}+\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} . \\ $$

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