Let $\{a_{n}\}$ be a sequence of real numbers. Construct an example for which $a_{n}\ge 0$ and $limsup_{n}na_{n}=\infty$ and yet $\sum_{n}a_{n}$ is convergent.
I found and understood the answer on this website for the case $limsup_{n}na_{n}>0$. But infinity is more extreme and I could not find an exact idea. I am very sorry for forgetting the most basic terms in the previous version of this question
For the old version:
There is no such sequence. If $\sum a_n$ is convergent then $a_n \to 0$. Any convergent sequence is bounded so $\lim \sup a_n <\infty$.
For the revised version of the question take $a_n=\frac 1{\sqrt n}$ if $n=m^{4}$ for some, $m$ and $0$ otherwise.