Identifying a power series I'm no analyst, so when a student in the class to whom I was teaching some elementary theory of (power) series, asked about this:
$\displaystyle{\sum_{n=0}^\infty\frac{x^n}{n^n}=1+x+\left(\frac{x}{2}\right)^2+\left(\frac{x}{3}\right)^3+\left(\frac{x}{4}\right)^4+\cdots}$
(assuming $0^0=1$), I had no idea.  It didn't look like anything I recognized, and a play about with some derivatives gave me no useful information.
Is this a known function, and does this series admit of an explicit formula?

To start analyzing this series, you could consider using Sterling's approximation $n!\approx \left(\frac ne\right)^n\sqrt {2\pi n}$:
$$\sum_{n=0}^\infty\left(\frac xn\right)^n\approx \sqrt{2\pi}\sum_{n=0}^\infty \frac {\sqrt n({x\over e})^n}{n!}\ge \sqrt{2\pi}\sum_{n=0}^\infty \frac {({x\over e})^n}{n!}=e^{\frac xe} \sqrt{2\pi}$$
Of course this is only a lower limit to the approximated value, and that $\sqrt n$ term still needs to be dealt with...
