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

On $[3,\infty )$, $h(x)=1/(x\log x)$ is positive, continuous and decreasing, since its denominator has positive derivative $\log x+1$. Substituting $u=\log x$ givesThe integral test for convergence therefore shows that the series is divergent.