# Calculate a 3rd order Maclaurin series phi( х ) = ( 1 + х)^(1 / х),

calculate a 3rd order Maclaurin series
$f\left(x\right)=\left(1+x{\right)}^{1/x},$
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Lorenzo Acosta
$\frac{\mathrm{ln}\left(1+x\right)}{x}=1-\frac{x}{2}+\frac{{x}^{2}}{3}-\frac{{x}^{3}}{4}+o\left({x}^{3}\right)$
Substitute $-\frac{x}{2}+\frac{{x}^{2}}{3}-\frac{{x}^{3}}{4}$ to $u$ in the development of ${\mathrm{e}}^{u}$, first computing the succesive powers of $u$
$\begin{array}{rl}{u}^{2}& =\frac{{x}^{2}}{4}-\frac{{x}^{3}}{3}+o\left({x}^{3}\right),\\ {u}^{3}& -\frac{{x}^{3}}{8}+o\left({x}^{3}\right),\end{array}$
so that
$\left(1+x{\right)}^{\frac{1}{x}}=\mathrm{e}\left(1+u+\frac{{u}^{2}}{2}+\frac{{u}^{3}}{6}+o\left(u\right)\right)=\mathrm{e}\left(1-\frac{x}{2}+\frac{11}{48}{x}^{2}-\frac{7}{16}{x}^{3}+o\left({x}^{3}\right)\right).$