Editing Series (mathematics) (section)

==Definition==

=== 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>

=== Partial sum of a series ===
Given a series <math display=inline>s=\sum_{k=0}^\infty a_k</math>, its {{tmath|n}}th ''partial sum'' is<ref name=":4" /><ref name=":2" /><ref name=":3" /><ref name=":15" />
<math display=block>s_n = \sum_{k=0}^{n} a_k = a_0 + a_1 + \cdots + a_n .</math>

Some authors directly identify a series with its sequence of partial sums.<ref name=":4" /><ref name=":3" /> Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,
<math display=block>a_n = s_{n} - s_{n-1}. </math>

Partial summation of a sequence is an example of a linear [[sequence transformation]], and it is also known as the [[prefix sum]] in [[computer science]]. The inverse transformation for recovering a sequence from its partial sums is the [[finite difference]], another linear sequence transformation. 

Partial sums of series sometimes have simpler closed form expressions, for instance an [[arithmetic series]] has partial sums
<math display=block>
s_n = \sum_{k=0}^{n} \left(a + kd\right)
= a + (a + d) + (a + 2d) + \cdots + (a + nd)
= (n+1)\bigl(a + \tfrac12 n d\bigr),
</math>
and a [[geometric series]] has partial sums<ref name=":45">{{harvnb|Spivak|2008|pp=473–478}}</ref><ref name=":24">{{harvnb|Apostol|1967|pp=388–390, 399-401}}</ref><ref name=":16">{{harvnb|Rudin|1976|p=61}}</ref>
<math display=block>s_n = \sum_{k=0}^{n} ar^k = a + ar + ar^2 + \cdots + ar^n  = a\frac{1 - r^{n+1}}{1 - r}</math>
if {{tmath|r \neq 1}} or simply {{tmath|1= s_n = a(n+1)}} if {{tmath|1= r = 1}}.

===Sum of a series===
[[File:Geometric sequences.svg|thumb|right|Illustration of 3 [[geometric series]] with partial sums from 1 to 6 terms. The dashed line represents the limit.]]
Strictly speaking, a series is said to [[Convergent series|''converge'']], to be ''convergent'', or to be ''summable'' when the sequence of its partial sums has a  [[Limit of a sequence|limit]]. When the limit of the sequence of partial sums  does not exist, the series [[Divergent series|''diverges'']] or is ''divergent''.<ref name=":43">{{harvnb|Spivak|2008|p=453}}</ref> When the limit of the partial sums exists, it is called the ''sum of the series'' or ''value of the series'':<ref name=":4" /><ref name=":2" /><ref name=":3" /><ref name=":15" />
<math display=block>\sum_{k = 0}^\infty a_k = \lim_{n\to\infty} \sum_{k=0}^n a_k = \lim_{n\to\infty} s_n.</math>
A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms.<ref>{{Cite journal |last=Knuth |first=Donald E. |year=1992 |title=Two Notes on Notation |journal=American Mathematical Monthly |volume=99 |issue=5 |pages=403–422 |doi=10.2307/2325085 |jstor=2325085 }}</ref> When the sum exists, the difference between the sum of a series and its <math>n</math>th partial sum, <math display=inline>s - s_n = \sum_{k=n+1}^\infty a_k,</math> is known as the <math>n</math>th ''[[truncation error]]'' of the infinite series.<ref name="Atkinson">{{Cite book |last=Atkinson |first=Kendall E. |title=An Introduction to Numerical Analysis |year=1989 |publisher=Wiley |isbn=978-0-471-62489-9 |edition=2nd |location=New York |page=20 |language=English |oclc=803318878}}</ref><ref name="Stoer">{{cite book |last1=Stoer |first1=Josef |title=Introduction to Numerical Analysis |year=2002 |edition=3rd |place=Princeton, N.J. |publisher=Recording for the Blind & Dyslexic |language=English |oclc=50556273 |last2=Bulirsch |first2=Roland}}</ref>

An example of a convergent series is the geometric series
<math display=block> 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8} + \cdots + \frac{1}{2^k} + \cdots.</math>

It can be shown by algebraic computation that each partial sum <math>s_n</math> is 
<math display=block>\sum_{k=0}^n \frac 1{2^k} = 2-\frac 1{2^n}.</math> As one has 
<math display=block>\lim_{n \to \infty} \left(2-\frac 1{2^n}\right) =2,</math>
the series is convergent and converges to {{tmath|2}} with truncation errors <math display=inline> 1 / 2^n </math>.<ref name=":45" /><ref name=":24" /><ref name=":16" />

By contrast, the geometric series
<math display=block>\sum_{k = 0}^\infty 2^k</math>
is divergent in the [[real number]]s.<ref name=":45" /><ref name=":24" /><ref name=":16" /> However, it is convergent in the [[extended real number line]], with <math>+\infty</math> as its limit and <math>+\infty</math> as its truncation error at every step.<ref>{{Cite web |last=Wilkins |first=David |year=2007 |title=Section 6: The Extended Real Number System |url=https://www.maths.tcd.ie/~dwilkins/Courses/221/Extended.pdf |access-date=2019-12-03 |website=maths.tcd.ie}}</ref>

When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly, [[convergence tests]] can be used to prove that the series converges or diverges.