TITLE: What is the series expansion of $ \sqrt[x] x $?
QUESTION [5 upvotes]: What is the series expansion of $ \sqrt[x] x $ ? I want to find it because I want to find $ \int \sqrt[x] x dx$ but believe that the integral cannot be expressed in the form of elementary functions.

REPLY [4 votes]: Easier than I thought.
We can write $ x ^ \frac {1}{x} $ as:
$$ x ^ \frac {1}{x}= e ^ { \frac {ln x} {x}} $$
The expansion of $e^x$ is:
$$ e^x=\sum_{n=0}^\infty \frac {x^n}{n!}$$ 
Similarly the expansion of $e ^ { \frac {ln x} {x}}$ is:
$$ e ^ { \frac {ln x} {x}}=\sum_{n=0}^\infty \frac {(\ln x)^n}{x^n.n!} $$