|
|
| Line 1: |
Line 1: |
| In [[mathematics]], an [[infinite geometric series]] of the form
| | Hi there, I am Alyson Boon even though it is not the title on my beginning certification. I've always loved residing in Kentucky but now I'm contemplating other options. Invoicing is what I do for a living but I've always wanted my personal business. To climb is some thing I truly appreciate performing.<br><br>My web site [http://165.132.39.93/xe/visitors/372912 best psychic readings] |
| :<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]]
| |
Hi there, I am Alyson Boon even though it is not the title on my beginning certification. I've always loved residing in Kentucky but now I'm contemplating other options. Invoicing is what I do for a living but I've always wanted my personal business. To climb is some thing I truly appreciate performing.
My web site best psychic readings