If $\limsup(na_n) = 1$, then $\sum\limits_{n=1}^{\infty} a_n$ diverges

Question

Let $a_n$ be a sequence of positive numbers. Suppose $\limsup(na_n) = 1$. Does this mean $\sum a_n$ diverges?

I have only concluded this if the limit superior is in fact the limit of the sequence. Also, I have managed to prove this if the limit inferior is strictly greater than $0$ (but then it is true for every limit superior)

Answer 1

No. Consider $a_n=1/n$ while $n=2^k$ for some $k$, and $a_n=1/n^2$ while $2^k<n<2^{k+1}$.