Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.
(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$
(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to \frac{1}{2}$
(c) $\sum_{k=1}^n \frac{k^2c_k}{n}$ converges.
Can we conclude any information about the sequence from it? It is tempting to guess $c_k \sim \frac{1}{2k^2}$ but that certainly requires a proof. Also, is it necessarily true that $\frac{c_n}{c_{n+1}}$ is bounded? Any suggestion regarding this is highly appreciated. It will be of great help even if one can provide some references for similar problems.