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f-x-1-n-2-x-n-n-




Question Number 135103 by frc2crc last updated on 10/Mar/21
f(x)=1+Σ_(n=2) ^∞ (((−x)^n )/n)
$${f}\left({x}\right)=\mathrm{1}+\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−{x}\right)^{{n}} }{{n}} \\ $$
Commented by Dwaipayan Shikari last updated on 10/Mar/21
1−(−(x^2 /2)+(x^3 /3)−..)  =1−(x−(x^2 /2)+(x^3 /3)−...)+x  =1+x+log(1+x)=log((e^(1+x) /(1+x)))
$$\mathrm{1}−\left(−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}−..\right) \\ $$$$=\mathrm{1}−\left({x}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}^{\mathrm{3}} }{\mathrm{3}}−…\right)+{x} \\ $$$$=\mathrm{1}+{x}+{log}\left(\mathrm{1}+{x}\right)={log}\left(\frac{{e}^{\mathrm{1}+{x}} }{\mathrm{1}+{x}}\right) \\ $$

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