Does $\sum_{i=1}^\infty a_i/i < \infty$ imply that $a_i$ has Cesaro mean zero? If $(a_i)_{i=1}^\infty$ is a sequence of positive real numbers such that: 
$$ \sum_{i=1}^\infty \frac{a_i}{i} < \infty. $$
Does this mean that the sequence $(a_i)_{i=1}^\infty$ has Cesaro mean zero? As in
$$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n a_i = 0.$$

Put $S_0=0$ and for $n\geq 1$, $\displaystyle S_n=\sum_{k=1}^n \frac{a_k}{k}$; note that $S_n$ is convergent, say to $S$. We have for $n\geq 1$ that $a_n=n(S_n-S_{n-1})$ and hence $a_1+\cdots+a_n=nS_n-\sum_{k=0}^{n-1} S_k$. Thus
$$T_n=\frac{a_1+\cdots +a_n}{n}=S_n-\frac{\sum_{k=1}^{n-1}S_k}{n}$$
and $T_n\to S-S=0$, and we are done.
