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let-f-x-n-1-x-n-1-x-n-with-x-0-1-prove-that-f-x-x-1-ln-1-x-x-1-




Question Number 30431 by abdo imad last updated on 22/Feb/18
let f(x)= Σ_(n=1) ^∞   (x^n /(1−x^n ))  with x∈[0,1[  prove that  f(x)∼_(x→1)    ((ln(1−x))/(x−1)).
$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{x}^{{n}} }{\mathrm{1}−{x}^{{n}} }\:\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\left[\:\:{prove}\:{that}\right.\right. \\ $$$${f}\left({x}\right)\sim_{{x}\rightarrow\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}−\mathrm{1}}. \\ $$

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