en>Christian75: /* Structure */Clean up, typos fixed: refelcting → reflecting, dimentional → dimensional using AWB

2013-03-31T20:31:09Z

Structure: Clean up, typos fixed: refelcting → reflecting, dimentional → dimensional using AWB

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{{Multiple image
|image1=Sum1111Plain.svg |alt1=A graph depicting the series with layered boxes |caption1=The series 1 + 1 + 1 + 1 + ⋯
|image2=Sum1111Smoothed.svg |alt2=A graph depicting the smoothed series with layered curving stripes |caption2=After smoothing
}}
[[File:Sum1111Asymptote.svg|thumb|Asymptotic behavior of the smoothing. The y-intercept of the line is −1/2.<ref>{{Citation |first=Terence |last=Tao |authorlink=Terence Tao |date=April 10, 2010 |title=The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation |url=http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ |accessdate=January 30, 2014}}</ref> |alt=A graph showing a line that dips just below the y-axis]]

In [[mathematics]], '''1 + 1 + 1 + 1 + · · ·''', also written <math>\sum_{n=1}^{\infin} n^0</math>, <math>\sum_{n=1}^{\infin} 1^n</math>, or simply <math>\sum_{n=1}^{\infin} 1</math>, is a [[divergent series]], meaning that its sequence of [[partial sum]]s do not converge to a [[limit of a sequence|limit]] in the [[real number]]s. The sequence 1''n'' can be thought of as a [[geometric series]] with the ratio 1. Unlike other geometric series with [[rational number|rational]] ratio (except −1), it neither converges in real numbers nor in [[p-adic number|{{mvar|p}}-adic numbers]] for some {{mvar|p}}. In the context of the [[extended real number line]]
: <math>\sum_{n=1}^{\infin} 1 = +\infty \, ,</math>
since its sequence of partial sums increases [[monotonic function|monotonically]] without bound.

Where the sum of {{math|''n''0}} occurs in [[theoretical physics|physical]] applications, it may sometimes be interpreted by [[zeta function regularization]]. It is the value at {{math|1=''s'' = 0}} of the [[Riemann zeta function]]
:<math>\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s}\,,</math>

The two formulas given above are not valid at zero however, so one must use the [[analytic continuation]] of the Riemann zeta functions,

:<math>
\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)
\!,</math>

Using this one gets (given that <math>\Gamma(1) = 1</math>),
:<math>
\zeta(0) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \sin\left(\frac{\pi s}{2}\right)\ \zeta(1-s) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \left( \frac{\pi s}{2} - \frac{\pi^3 s^3}{48} + ... \right)\ \left( -\frac{1}{s} + ... \right) = -\frac{1}{2}
\!</math>
where the power series expansion for {{math|ζ(''s'')}} about {{math|1=''s'' = 1}} follows because {{math|ζ(''s'')}} has a simple pole of [[residue (complex analysis)|residue]] one there. In this sense {{math|1=1 + 1 + 1 + 1 + · · · = ζ(0) = −1⁄2}}.

[[Emilio Elizalde]] presents an anecdote on attitudes toward the series:
{{blockquote|1=In a short period of less than a year, two distinguished physicists, [[Andrei Slavnov|A. Slavnov]] and [[Francisco José Ynduráin|F. Yndurain]], gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, {{math|1=1 + 1 + 1 + · · · = −1⁄2}}'. Implying maybe: ''If you do not know this, it is no use to continue listening.''<ref>{{cite conference |first=Emilio |last=Elizalde |title=Cosmology: Techniques and Applications |booktitle=Proceedings of the II International Conference on Fundamental Interactions |year=2004 |arxiv=gr-qc/0409076}}</ref>}}

==See also==
* [[1 − 1 + 2 − 6 + 24 − 120 + · · ·]]
* [[Harmonic series (mathematics)|Harmonic series]]

==Notes==
{{reflist}}

==External links==
*{{OEIS|A000027}}

{{Series (mathematics)}}

{{DEFAULTSORT:1 + 1 + 1 + 1 + ...}}
[[Category:Arithmetic series]]
[[Category:Divergent series]]
[[Category:Geometric series]]
[[Category:One]]
[[Category:Mathematics paradoxes]]

Polyiodide - Revision history

en>Christian75: /* Structure */Clean up, typos fixed: refelcting → reflecting, dimentional → dimensional using AWB