Ramanujan's expansion of $\operatorname{li}(x)$

Question

What are the steps to showing

$$ \operatorname{li}(x)=\gamma+\log(\log(x))+\sum_{k=1}^{\infty}\dfrac{\log(x)^k}{k!k}? $$

Any pointers on where to look would be warmly appreciated.

Answer 1

Hint: If you differentiate the RHS you get $$\frac{1}{\log x} \frac{1}{x}+\sum_{k=1}^\infty \frac{1}{x} \frac{\log (x)^{k-1}}{k!}=\frac{1}{\log x} \frac{1}{x}+ \frac{1}{x \log x} \left(e^{\log x}-1 \right)=\frac{1}{\log x} .$$