I have a conjecture regarding series convergence that feels like it would be a useful tool to me if I could prove it, but I have been unable to prove or disprove it.
Let $\sum a_n$ be a series of nonnegative terms. Define $\lambda(N)$ to be the number of terms of the series that are greater than $1/N$.
Conjecture: If $\lambda(N)=O(N^\alpha)$, with $0<\alpha<1$, then $\sum a_n$ converges.
The conjecture is based on the fact that this is true for series of the form $\sum 1/n^k$, $k$ constant (let $\alpha = 1/k$), and my intuition that the hypothesis of the conjecture is enough to make $\sum a_n$ "sufficiently similar to or bounded by" such a series.
Can you offer a counterexample or point me toward an idea for a proof?
(I tried to bound $\Delta\lambda(N) = \lambda(N+1)-\lambda(N)$ from the assumption $\lambda(N)=O(N^\alpha)$, since possibly excluding a finite number of terms, the series sum is bounded above by $\sum \Delta\lambda(N)/N$; but I couldn't see how to do this.)