= Solution
A real <sequence> converges to a finite limit $L$ if
$$
\forall\varepsilon>0\ \exists N\ \forall n\geq N:\quad |x_n-L|<\varepsilon.
$$
A <series> converges when its <partial sums> $s_N=\sum_{n=1}^N x_n$ form a <convergent sequence> with a finite limit. These are different conditions: a <series> concerns accumulated terms, whereas <sequence> convergence concerns the terms themselves.
If $s_N\to S$, then $s_{N-1}\to S$ as well and $x_N=s_N-s_{N-1}\to0$. Explicitly, choose $N$ large enough that both <partial sums> lie within $\varepsilon/2$ of $S$; the <triangle inequality> gives $|x_N|<\varepsilon$. Hence \b[convergence of a series forces its terms to tend to zero].
For the converse, take $x_n=1/n$. This is a <null sequence>, but its <harmonic series> diverges. In each block $2^{j-1}<n\leq2^j$, there are $2^{j-1}$ terms each at least $2^{-j}$, so the block contributes at least $1/2$. Thus
$$
s_{2^k}\geq1+\frac k2\longrightarrow\infty.
$$
\b[A zero limit for the terms is necessary, but not sufficient, for <convergent series>.]
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