How does Cesaro summability imply the partial sums converge to the same sum? I can't reconcile this fact I used to know.
Suppose you have a sequence of nonnegative terms $a_k$. Let $s_n=\sum_{k=1}^n a_k$, and suppose
$$
\lim_{n\to\infty}\frac{s_1+\cdots+s_n}{n}=L.
$$
Then $\sum_{k=1}^\infty a_k$ also exists and equals $L$.
I could recover that $\sum_{k=1}^\infty a_k$ exists. If not, it diverges to $\infty$. Suppose $M>0$ is given. There exists $N$ such that $\sum_{k=1}^N a_k>M$. If $n>N$, then
$$
\begin{align*}
\frac{s_1+\cdots+s_n}{n} &=\frac{s_1+\cdots+s_N}{n}+\frac{s_{N+1}+\cdots+s_n}{n}\\
&\geq\frac{n-N}{n}M.
\end{align*}
$$
Taking $n\to\infty$ shows that $L\geq M$ for all positive $M$, which is clearly not true. 
But I can't for the life of me remember why $\sum_{k=1}^\infty a_k=L$ and can't find it online. Can someone clear this up for me? Thanks.

No this is not true. $a_k = (-1)^k$ is a counterexample.
