Quinhydrone electrode

2013-12-05T01:51:07Z

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In [[mathematics]], an [[infinite geometric series]] of the form
:<math>\sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots</math>
is [[divergent series|divergent]] if and only if | ''r'' | ≥ [[1 (number)|1]]. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case
:<math>\sum_{k=0}^\infty ar^k = \frac{a}{1-r}</math>.
This is true of any summation method that possesses the [[Divergent_series#Properties_of_summation_methods|properties]] of [[regularity]]{{disambiguation needed|date=May 2012}}, [[linearity]], and [[Stability theory|stability]].{{Disambiguation needed|date=August 2011}}

==Examples==
In increasing order of difficulty to sum:
*[[1 − 1 + 1 − 1 + · · ·]], whose common ratio is [[−1 (number)|−1]]
*[[1 − 2 + 4 − 8 + · · ·]], whose common ratio is −2
*[[1 + 2 + 4 + 8 + · · ·]], whose common ratio is [[2 (number)|2]]
*[[1 + 1 + 1 + 1 + · · ·]], whose common ratio is 1.

==Motivation for study==
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called '''Borel-Okada principle''': If a [[regular summation method]] sums Σ''z''<sup>''n''</sup> to 1/(1 - ''z'') for all ''z'' in a subset ''S'' of the [[complex plane]], given certain restrictions on ''S'', then the method also gives the [[analytic continuation]] of any other function {{nowrap|1=''f''(''z'') = Σ''a''<sub>''n''</sub>''z''<sup>''n''</sup>}} on the intersection of ''S'' with the [[Mittag-Leffler star]] for ''f''.<ref>Korevaar p.288</ref>

==Summability by region==

===Open unit disk===
Ordinary summation succeeds only for common ratios |''z''| < 1.

===Closed unit disk===
*[[Cesàro summation]]
*[[Abel summation]]

===Larger disks===
*[[Euler summation]]

===Half-plane===
The series is [[Borel summation|Borel summable]] for every ''z'' with real part < 1. Any such series is also summable by the generalized Euler method (E, ''a'') for appropriate ''a''.

===Shadowed plane===
Certain [[moment constant method]]s besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 − ''z''), that is, for all ''z'' except the ray ''z'' ≥ 1.<ref>Moroz p.21</ref>

===Everywhere===

==Notes==
{{reflist}}

==References==
<div class="references-small">
*{{cite book |last=Korevaar |first=Jacob |title=Tauberian Theory: A Century of Developments |publisher=Springer |year=2004 |isbn=3-540-21058-X}}
*{{cite arxiv |first=Alexander |last=Moroz |title=Quantum Field Theory as a Problem of Resummation |year=1991 |eprint=hep-th/9206074 }}
</div>

{{Series (mathematics)}}

[[Category:Divergent series]]
[[Category:Geometric series]]

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