# A Closed form for series over Exponential Integral

Tags:
1. Feb 16, 2017

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?

Last edited: Feb 16, 2017
2. Feb 23, 2017