Maclaurin series for a function Provided I have the function 
\begin{equation*}
f(x)=(1+x)^{1/x}, 
\end{equation*}
and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as this seems hard..). I know that
\begin{equation*}
(1+x)^{1/x}=e^{ln(1+x)/x}, 
\end{equation*}
and the Maclaurin series for $e^x$ is easy to prove, so I think it's a good direction..

$$\frac{\ln(1+x)}x=1-\frac x2+\frac{x^2}3-\frac{x^3}4+o(x^3)$$
Substitute $-\frac x2+\frac{x^2}3-\frac{x^3}4$ to $u$ in the development of $\mathrm e^u$, first computing the succesive powers of $u$:
\begin{align*}
u^2&=\frac{x^2}4-\frac{x^3}3+o(x^3),\\
u^3&-\frac{x^3}8+o(x^3),
\end{align*}
so that $$(1+x)^{\tfrac 1x}=\mathrm e\Bigl(1+u+\frac{u^2}2+\frac{u^3}6+o(u)\Bigr)=\mathrm e\Bigl(1-\frac x2+\frac{11}{48}x^2-\frac 7{16}x^3+o(x^3)\Bigr).$$
