$\displaystyle \sum_{1}^{\infty}\frac{ln k}{k\sqrt {k}}$
how to shpw if it converges or diverges
For $\displaystyle \text{Re}\ s>1 $ the 'Riemann zeta function' is defined as...
$\displaystyle \zeta(s)= \sum_{k=1}^{\infty} \frac{1}{k^{s}}$ (1)
... and its derivative is...
$\displaystyle \zeta^{'}(s)= - \sum_{k=1}^{\infty} \frac{\ln k}{k^{s}}$ (2)
... so that is...
$\displaystyle \sum_{k=1}^{\infty} \frac{\ln k}{k^{\frac{3}{2}}}= - \zeta^{'}(\frac{3}{2})$ (3)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$