Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 1 / 10F / ii / Solution

Because $3>e$, each of $x$, $\log x$ and $\log \log x$ is positive and increasing for $x\ge 3$. Hence $h(x)=1/[x\log x(\log \log x{)}^2]$ is positive, continuous and decreasing. With $u=\log \log x$, $du=dx/(x\log x)$, soThe integral test for convergence implies that the series is convergent.