Necessary and sufficient condition for convergent series

Question

Let $(a_i)_{i \in \mathbb{N}}$ be a sequence of positive reals such that $$ \limsup_{i \rightarrow \infty} a_i \, i =0. $$ Is this condition necessary and sufficient for $\sum\limits_{i=1}^\infty a_i < \infty$?

Of course, if $a_i = 1/i$, then the series is infinite and if $a_i = (1/i)^{1 + \varepsilon}$, for some $\epsilon >0$, then it is finite, but it is not clear to me what happens if I choose a sequence which decays faster than $1/i$ but not faster than $(1/i)^{1 + \varepsilon}$ for arbitrary small $\varepsilon>0$.

Answer 1

Note that $\sum_{n=2}^\infty\frac1{n\log n}$ diverges (this follows from the integral test), but $\lim_{n\to\infty}n\frac1{n\log n}=0$.

Answer 2

It's obviously necessary: if $\sum_na_n<\infty$, then $a_n\to0$, so $a_n/n\to0$.

It is really far from sufficient: take $a_n=\sqrt{n}$; then $a_n/n\to0$, while the series diverges spectacularly.