The question is pretty simple:
It the series $\sum \frac{(-1)^{\lfloor ne\rfloor}}{n}$ convergent or not ?
As usual in this situation we let $S_n:=\sum_{k=0}^n(-1)^{\lfloor ne\rfloor}$ and apply an Abel transform to the original series. Thus the convergence of the original series is equivalent to that of $\sum \frac{S_n}{n^2}$.
We have the obvious bound $S_n = O(n)$. However in order to conclude one might need to achieve a bound of the form $S_n=O(n^{\alpha})$ for some $\alpha <1$. Does anyone know a proof of this fact? (Or a proof for the original quesiton).