Is there a closed form for the constant given by:
$$\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(2))}{n}$$
(Where $Ei$ is the exponential integral)?
Could we generalize it for:
$$I(k)=\sum_{n=2}^\infty \frac{Ei(-(n-1)\log(k))}{n}$$
?
My try: As it is given that $k$ will be a positive integer, I have already proved that these series converge at least for every $k>1$. To obtain a closed form, I have tried to substitute the exponential integral by both its main definition and its series expansion, with no success. Mathematica does not give any result either. Any help?