= Series
{disambiguate=mathematics}
{title2=$\sum_{n=1}^{\infty}a_n$}
{wiki}
= Series
{synonym}
A <series> adds the terms of a <sequence> in their specified order. Its infinite-sum notation is defined through finite <partial sums>:
$$
S_N=\sum_{n=1}^{N}a_n,\qquad
\sum_{n=1}^{\infty}a_n=\lim_{N\to\infty}S_N
$$
when the displayed <limit of a sequence> exists as a finite number. Otherwise the <series> is not a <convergent series> in the ordinary sense. Writing an infinite list of additions does not by itself assign it a finite sum. <Cesàro summation> is a different prescription, with its own definition.
Two positive <series> make the distinction concrete. For $a_n=1/n$, the individual terms tend to zero but the <harmonic series> has unbounded <partial sums>. For $a_n=1/[n(n+1)]$, <partial fraction decomposition> reveals a <telescoping series> whose <partial sums> tend to one. The useful question is how much all the remaining terms can contribute together, rather than only how small the next term looks.
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