**Question:** Let $a\_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if


$$\left \lvert \frac{1}{N} \left ( \sum\_{n=0}^{N-1} a\_n \right )\right \rvert < \frac{1}{N^{1+\varepsilon}}$$


for all $N \in \mathbb Z\_+$, then $a\_n$ converges to $0$?



Does rapid convergence of the Cesaro sums imply convergence of the original sequence?
Multiply by $N$ to get that


$$\left \lvert \sum\_{n=0}^{N-1} a\_n \right \rvert < \frac{1}{N^{\varepsilon}}$$


Therefore $\sum\_{n=0}^\infty a\_n=0$ and $\lim\_{n\to\infty} a\_n=0$.


