$\sum_{n=0}^{\infty}a_n$ diverges in the regular term but is Cesaro summable
Prove $a_n/n\to 0$ when $n\to \infty$
We used the definition of the Cesaro sum and obtained:
$\lim_{N\to \infty}\frac{1}{N+1}\sum_{n=0}^{N}S_n$=$\lim_{N\to \infty}\frac{1}{N+1}\left ( \sum_{n=0}^{N-1}S_n+\sum_{k=0}^{N}a_k\right )$=$\sum_{n=0}^{N-1}\frac{S_n}{N+1}$$+\sum_{k=0}^{N}\frac{a_k}{N+1}$=$L$
but from here we're kind of stuck
*($S_n$ is the partial sums of $a_n$)