Consider the sum $\sum_{x=1}^{\infty} \frac{\log{x}}{z^x}$. We can assume that $z\geq1$ (and is real). Mathematica gives this sum as
-PolyLog^(1, 0)[0,1/z]
Even after reading the manual page for PolyLog I don't understand what this function is like and I certainly don't know how the sum was derived.
Are there simple upper and lower bounds for this sum?
I also tried to compute $\int_{x=1}^{\infty} \frac{\log{x}}{z^x}$ in the hope that this would shed more light but Mathematica gives
Gamma[0, Log[z]]/Log[z]
which I also didn't find helpful.