The problem:
How to find the sum?
$-\sum_{i=1}^{\infty}\frac{(-x)^{i\; \bmod(k-1)}}{i}$
Details:
I tried find this sum using Mathematica
-Sum[((-x)^(Mod[i,k-1]))/i,{i,1,Infinity}]
but I got no answer.
This sum is based on the $\ln(x+1)=-\displaystyle\sum_{i=1}^{\infty}\frac{(-x)^i}{i}$
The final result that I need is formula with finite indices, something like $-\displaystyle\sum_{j=0}^{k-2}f(x,k)$, so how to find this $f(x,k)$?
This another question shows exactly what I need.
Unfortunally I could't find how to multisect this function and find this sum until now.