How would I find a closed form of $ \sum_{n=1}^{\infty} \frac{x^{\ln n}}{n!} $?
By the ratio test, the series converges for all x:
$$ \underset{n \to \infty}{\lim} \left| \frac{x^{\ln (n+1)}n!}{x^{\ln n}(n+1)!} \right| = \underset{n \to \infty}{\lim} \left| \frac{x^{\ln(1+n^{-1})}}{n+1} \right| = 0$$
I've tried differentiating and integrating the series, but to no luck.