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Question Number 38725 by maxmathsup by imad last updated on 28/Jun/18
let f(x)=ln(1+ e^(−x) ) developp f at integr serie .
letf(x)=ln(1+e−x)developpfatintegrserie.
Commented by abdo.msup.com last updated on 29/Jun/18
we have ln(1+u) =Σ_(n=1) ^∞ (((−1)^(n−1) )/n) x^n ⇒ ln(1+e^(−x) ) =Σ_(n=1) ^∞ (((−1)^n )/n) e^(−nx) =Σ_(n=1) ^∞ (((−1)^n )/(n!)) Σ_(p=0) ^∞ (((−nx)^p )/(p!)) =Σ_(n=1) ^∞ (((−1)^n )/(n!)){Σ_(p=0) ^∞ (((−n)^p )/(p!))x^p }.
wehaveln(1+u)=∑n=1∞(−1)n−1nxn⇒ln(1+e−x)=∑n=1∞(−1)nne−nx=∑n=1∞(−1)nn!∑p=0∞(−nx)pp!=∑n=1∞(−1)nn!{∑p=0∞(−n)pp!xp}.

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