Proof that series diverges Prove that $\displaystyle\sum_{n=1}^\infty\frac{1}{n(1+1/2+\cdots+1/n)}$ diverges.
I think the only way to prove this is to find another series to compare using the comparison or limit tests. So far, I have been unable to find such a series.

Hints: 1. For every $k\geqslant0$, the sum of $\frac1n$ from $n=2^k$ to $n=2^{k+1}-1$ is at most $1$. 2. For every $k\geqslant0$ and every $n$ such that $2^k\leqslant n\leqslant 2^{k+1}-1$,  $1+\frac12+\cdots+\frac1n\leqslant k+1$ and $\frac1n\geqslant\frac1{2^k}$. 3. For every $k\geqslant0$, the terms from $n=2^k$ to $n=2^{k+1}-1$ of the series which interests you sum to at least $\frac1{k+1}$. 4. The harmonic series diverges.
