Chapter 9.5, Problem 53ES

Show that $\ln x<\sqrt{x}\text{if}x>0,$ and use this result to investigate the convergence of

$\begin{array}{cc}\left(\text{a}\right){\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{\ln k}{k^2}}& \left(\text{b}\right){\displaystyle \underset{k=2}{\overset{\infty }{\sum }}\frac{1}{{\left(\ln k\right)}^2}}\end{array}$