Value of $\sum\limits^{\infty}_{n=1}\frac{\ln n}{n^{1/2}\cdot 2^n}$ Here is a series:
$$\displaystyle \sum^{\infty}_{n=1}\dfrac{\ln n}{n^{\frac12}\cdot 2^n}$$
It is convergent by d'Alembert's law. Can we find the sum of this series ?

Consider 
$$f(s):=\sum_{n=1}^\infty \frac {\left(\frac 12\right)^n}{n^s}=\operatorname{Li}_s\left(\frac 12\right)$$
with $\operatorname{Li}$ the polylogarithm then (since $\,n^{-s}=e^{-s\ln(n)}$) :
$$f'(s)=\frac d{ds}\operatorname{Li}_s\left(\frac 12\right)=-\sum_{n=1}^\infty \frac {\ln(n)}{n^s}\left(\frac 12\right)^n$$
giving minus your answer for $s=\frac 12$.
You may use the integrals defining the polylogarithm to get alternative formulations but don't hope much simpler expressions...
