Just something that's been in my head for a while:
If $a_n$ is a sequence of positive real numbers for $n\in{1,2,...,N}$ and it is known that $$ S_\epsilon(N)=\sum_{k=1}^{N}\frac{a_k}{k^{1+\epsilon}} $$ $$ \lim_{N\rightarrow\infty} S_\epsilon(N) < \infty $$ for all $\epsilon>0$ and $S_0(N)=\infty$, is there anything more we can gain from this information other than $$ \lim_{k\rightarrow\infty}\frac{a_k}{k^{1+\epsilon}}=0 ? $$
It just seems like there's more information to be gained, but it seems like even making the statement $$ \lim_{k\rightarrow\infty}\frac{a_k}{k}=0 $$
isn't necessarily the case, but am I mistaken? It seems like we know something more about the boundary of $\epsilon$ parameter.