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.

This answer is similar in spirit to Didier Piau's answer.
The following theorem is a very useful tool:
Suppose that $a_k > 0$ form a decreasing sequence of real numbers. Then
$$\sum_{k=1}^\infty a_k$$ converges if and only if $$\sum_{k=1}^\infty 2^k a_{2^k}$$ converges.
Applying this to the problem in hand we are reduced to investigating the convergence of
$$\sum_{k=1}^\infty \frac{1}{1 + 1/2 + \dots + 1/2^k}$$
But one easily sees that
$$1 + 1/2 + \dots + 1/2^k \le 1 + 2 \cdot 1/2 + 4 \cdot 1/4 + \dots + 2^{k-1}\cdot1/2^{k-1} + 1/2^k \le k + 1.$$
Because
$$\sum_{k=1}^\infty \frac{1}{k+1}$$
diverges, we are done.
