Closed form for series over Exponential Integral

[pkmpad]

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?

 

[jvicens]

Unfortunately, I am not aware of a closed form for the constant given by the first sum, nor have I been able to find a closed form for the generalized sum involving the variable k. The exponential integral function does not have a simple closed form, and it is difficult to manipulate in series form. It is possible that a closed form exists for these sums, but it may require advanced mathematical techniques to find it.

However, we can still make some observations about the behavior of these sums. As you mentioned, the series converges for all positive integer values of k. This is because as n approaches infinity, the term in the sum becomes smaller and smaller, approaching 0. Additionally, the behavior of the sum will depend on the value of k. For example, when k is close to 1, the sum will approach a finite value, but as k gets larger, the sum will approach infinity.

In terms of generalizing the sum, I would suggest exploring different values of k and seeing if any patterns emerge. You could also try manipulating the series in different ways, such as rearranging the terms or using different properties of the exponential integral function. However, finding a closed form may require more advanced mathematical techniques and may not be possible in this case.