Finding sum of the infinite series.

Question

Find the sum of the series $$\sum_{n=0}^{\infty} \left ( \frac {\log (\log x)} {n!} \right )^n.$$

How do I proceed? Please help me in this regard.

Thank you very much.

Answer 1

There is no closed form expression to this series. The general term is extremely quickly decreasing. With $t=\log(\log(x))$,

$$1+t+\frac{t^2}4+\frac{t^3}{216}+\frac{t^4}{331776}+\frac{t^5}{24883200000}+\cdots$$