Editing Series (mathematics) (section)

=== Series ===
A ''series'' or, redundantly, an ''infinite series'', is an infinite sum. It is often represented as<ref name=":5" /><ref>{{Cite book |last=Swokoski |first=Earl W. |title=Calculus with Analytic Geometry |publisher=Prindle, Weber & Schmidt |year=1983 |isbn=978-0-87150-341-1 |edition=Alternate |location=Boston |pages=501}}</ref><ref name=":15">{{harvnb|Rudin|1976|p=59}}</ref>
<math display=block>a_0 + a_1 + a_2 + \cdots \quad \text{or} \quad a_1 + a_2 + a_3 + \cdots, </math>
where the [[summand|terms]] <math>a_k</math> are the members of a [[sequence]] of  [[number]]s, [[Function (mathematics)|functions]], or anything else that can be [[addition|added]]. A series may also be represented with [[capital-sigma notation]]:<ref name=":5" /><ref name=":15" />
<math display=block>\sum_{k=0}^{\infty} a_k \qquad \text{or} \qquad \sum_{k=1}^{\infty} a_k . </math>

It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the {{tmath|n}}th term as a [[function (mathematics)|function]] of {{tmath|n}}:
<math display=block>a_0 + a_1 + a_2 + \cdots + a_n +\cdots \quad \text{ or } \quad f(0) + f(1) + f(2) + \cdots + f(n) + \cdots. </math>
For example, [[Euler's number]] can be defined with the series
<math display=block>\sum_{n=0}^\infty \frac 1{n!}=1+1+\frac12 +\frac 16 +\cdots + \frac 1{n!}+\cdots, </math> 
where <math>n!</math> denotes the product of the <math>n</math> first [[positive integer]]s, and <math>0!</math> is conventionally equal to <math>1.</math><ref name=":42">{{harvnb|Spivak|2008|p=426}}</ref><ref name=":22">{{harvnb|Apostol|1967|p=281}}</ref><ref>{{harvnb|Rudin|1976|p=63}}</ref>