Maclaurin series of (1+x)^(1/x)

Question

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$?

$f(0)$ is not even defined, or should I define it as $f(0)=e$?

I stopped at the first derivative as it gets terribly messy.

thank you

Answer 1

Hint

Start with $$\log(f)=\frac{1}{x} \log(1+x)=\frac{1}{x}\Big(x-\frac{x^2}{2}+\frac{x^3}{3}+O\left(x^4\right)\Big)= 1-\frac{x}{2}+\frac{x^2}{3}+O\left(x^3\right)$$ Now $$f=e^{\log(f)}=e^{1-\frac{x}{2}+\frac{x^2}{3}+O\left(x^3\right)}=e ~~ e^{-\frac{x}{2}+\frac{x^2}{3}+O\left(x^3\right)}$$ Now, set $y=-\frac{x}{2}+\frac{x^2}{3}$ and use $e^y=1+y+\frac{y^2}{2}+...$ Replace $y$ by its expression as a function of $x$ and develop. You will end with

$$f=e-\frac{e x}{2}+\frac{11 e x^2}{24}+O\left(x^3\right)$$