 Solution

The summand is positive, so it suffices to show that its partial sums are unbounded. On the block $2^k\le n<2^{k+1}$, for $k\ge 1$,There are $2^k$ terms in that block, so its sum is at least $1/[2(k+1)\log 2]$. Adding blocks gives a constant multiple of the divergent harmonic series. HenceEquivalently the integral test for convergence gives ${\int }_2^{R}dx/(x\log x)=\log \log R-\log \log 2\to \infty$. The block proof supplies the divergence without assuming the integral test.