Given $a_{n}\ge a_{n+1}>0$, how to prove $a_{n}=o(\frac{1}{n})$ is necessary for $\sum_{n=1}^{\infty}a_n$ to converge?
If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$