An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$.

Question

I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.

I have a really bad attempt:

Define $a_{0}=a_{1}=0$, and $a_{k}:=\log(\log(k))$ for $k\geq 2$, then $a_{k}\longrightarrow\infty$. But $$\dfrac{1}{n}\sum_{k=0}^{n-1}a_{k}=\dfrac{\log\Big(\prod_{k=2}^{n-1}\log(k)\Big)}{n}$$ and Wolframalpha told me that this series converges to $0$ as $n\longrightarrow\infty$.

However, I have no idea about how to show this convergence to $0$. Also, I wish to have a sequence as simple as possible, since at some stage I need too how $f(k):=a_{k}$ is positive semi-definite.

Is there any simpler example?

Thank you!

Answer 1

How about $a_k=(-1)^k$ C-sum $\to 0$, but $a_k$ does not converge.

Answer 2

Try $$a_n = \begin{cases} 1 & \text{$n$ is a power of $2$} \\ 0 & \text{otherwise}\end{cases}$$