(AMM. Problems and Solutions. 2015. 03) Let $sed{a_n}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $$ex vsm{n}a_n<infty eex$$ if and only if $a_n=O(1/ln n)$ and $dps{vsm{n}(a_n-a_{n+1}) ln n<infty}$.
必要性好办, 但是充分条件怎么办呢?