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