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.

Let $H_n = 1 + \frac{1}{2} + \ldots + \frac{1}{n}$. $H_n$ is known as harmonic sum. Note that $H_{n+1} - H_n = \frac{1}{n+1}$. Now, consider $d_n = H_n - 1 - \ln(n)$. Then $d_{n+1} - d_n = \frac{1}{n+1} + \ln\left( 1 - \frac{1}{n+1} \right) < 0$,  $\forall n \ge 1$ which follows from $\ln(1-x) < -x$, for $0<x<1$. 
Now, since $d_{n+1} - d_n < 0$ and $d_1 = 0$, it follows that $H_n \le 1 + \ln(n)$, and thus 
$\sum_{n=1}^m \frac{1}{n H_n} \ge \sum_{n=1}^m \frac{1}{n( 1 + \ln(n))}$. Since the latter diverges by integral test, so does the original sum.
