View source for Series (summation)

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==Definition==
Let {{M|(X,d)}} be a [[metric space]] such that {{M|(X,+)}} is a [[group]]. Let {{M|(x_n)_{n\in\mathbb{N} }\subseteq X}} be any [[sequence]] in {{M|X}}. Then:
* {{M|\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)}} is a "''series with general term {{M|x_n}}''"{{rAPIKM}} where:
** for all {{M|n\in\mathbb{N} }} we may define {{M|1=S_n:=\sum^n_{i=1}x_i}}, this is called the {{M|n}}<sup>th</sup> ''partial sum''{{rAPIKM}} of the series
If {{M|(s_n)_{n\in\mathbb{N} } }} [[convergent sequence|converges]] (in the usual sense for {{plural|sequence|s}}) then we may say:
* The series, {{M|\big((x_n)_{n\in\mathbb{N} },(S_n)_{n\in\mathbb{N} }\big)}}, converges{{rAPIKM}}.
This may be written:
* {{M|\sum^\infty_{n\eq 1}x_n<\infty}}{{rAPIKM}} but this is an abuse of notation.
==Work==
{{Requires work|grade=A|msg=Working on it right now}}

==References==
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{{Definition|Functional Analysis|Analysis|Real Analysis|Metric Space|Topology}}