Editing Series (mathematics) (section)

===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.