example of integral test

Consider the series

$$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k\log k}.$$

Since the integral

$${\int }_1^{\mathrm{\infty }}\frac{1}{x\log x}𝑑x=\underset{M\to \mathrm{\infty }}{lim}{\left[\log (\log (x))\right]}_1^M$$

is divergent also the series considered is divergent.