Q:

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges? 

A:

We can prove more generally that if $\beta_n\ge0$ and $\sum_n\beta_n$ diverges (in your case $\beta_n=n^{-1}$), then there exists $\alpha_n$ with the given properties such that $\sum_n\beta_n\alpha_n$ diverges. To construct such $\alpha_n$, given $\beta_n$, construct a (strictly) increasing sequence $n_k$ such that $\sum_{j=n_{k-1}}^{n_k}\beta_j\gt1$. (This is possible since $\sum_n\beta_n$ diverges.) Then let $\alpha_n=k_n^{-1}$, where $k_n$ is the unique integer with $n_{k_n}\le n\le n_{k_n+1}$. Since $k_n\to\infty$ with $n\to\infty$, we have $\alpha_n\to0$, as required. But $\sum_n\alpha_n$ and $\sum_n\beta_n\alpha_n$ diverge, the first because $k_n^{-1}\ge n^{-1}$, the second because $\sum_{n=1}^{n_m}\beta_n\alpha_n\gt\sum_{k=1}^mk^{-1}$.  
For more on this sort of argument and the lack of a boundary between convergent and divergent sequences, see this answer, and also this question.

