= Solution
A <sequence> $(a_k)$ converges to $a\in\mathbb R$ when for every $\varepsilon>0$ there is $N$ such that $|a_k-a|<\varepsilon$ for all $k\ge N$. A <series> $\sum_{k=1}^{\infty}a_k$ converges to a finite $S$ when its <partial sums> $s_n=\sum_{k=1}^n a_k$ converge to $S$.
If the <series> converges, then $a_n=s_n-s_{n-1}\to S-S=0$, so \b[its terms tend to zero]. The converse is false: the <harmonic series> has terms tending to zero but diverges.
For the arithmetic averages, suppose $a_k\to a$. Fix $\varepsilon>0$ and choose $K$ so $|a_k-a|<\varepsilon/2$ for $k\ge K$. The finite initial contribution $C=\sum_{k=1}^{K-1}|a_k-a|$ is fixed, and
$$
\left|\frac1n\sum_{k=1}^na_k-a\right|\le\frac Cn+\frac1n\sum_{k=K}^n|a_k-a|\le\frac Cn+\frac\varepsilon2.
$$
For sufficiently large $n$, $C/n<\varepsilon/2$, proving
$$
\boxed{\frac1n\sum_{k=1}^na_k\longrightarrow a.}
$$
This is the <Cesaro theorem for convergent sequences>. Its hypothesis concerns a <convergent sequence>; the corresponding <series> need not be a <convergent series>.
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