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Question Number 181570 by SANOGO last updated on 26/Nov/22

prove that:xε]−1,1[ Σ_(n=1) ^(+oo) (x^n /n)=−ln(1−x)

provethat:xϵ]−1,1[∑+oon=1xnn=−ln(1−x)

Answered by mr W last updated on 27/Nov/22

for −1<x<1: 1+x+x^2 +...=Σ_(n=1) ^∞ x^(n−1) =(1/(1−x)) Σ_(n=1) ^∞ ∫_0 ^x x^(n−1) dx=∫_0 ^x (1/(1−x))dx Σ_(n=1) ^∞ (x^n /n)=−ln (1−x) ✓

for−1<x<1:1+x+x2+...=∑∞n=1xn−1=11−x∑∞n=1∫0xxn−1dx=∫0x11−xdx∑∞n=1xnn=−ln(1−x)✓

Commented by SANOGO last updated on 27/Nov/22

thank you

thankyou

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