MRB constant: Difference between revisions
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{{Short description|Mathematical constant described by Marvin Ray Burns}} |
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[[File:MRB-Gif.gif|thumb|right|195px|First 100 partial sums of <math>(-1)^k (k^{1/k} - 1)</math>]]The '''MRB constant |
[[File:MRB-Gif.gif|thumb|right|195px|First 100 partial sums of <math>(-1)^k (k^{1/k} - 1)</math>]]The '''MRB constant''' is a [[mathematical constant]], with decimal expansion {{nowrap|0.187859…}} {{OEIS|A037077}}. The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999.<ref>{{cite web|url=http://www.plouffe.fr/simon/constants/mrburns.txt|title=mrburns|last=Plouffe|first=Simon|access-date=12 January 2015}}</ref> Burns had initially called the constant "rc" for root constant<ref>{{cite web|url=http://math2.org/mmb/thread/901|title=RC|last=Burns|first=Marvin R.|date=23 January 1999|website=math2.org|access-date=5 May 2009}}</ref> but, at [[Simon Plouffe|Simon Plouffe's]] suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant".<ref>{{cite web|url=http://www.plouffe.fr/simon/articles/Tableofconstants.pdf|title=Tables of Constants|last=Plouffe|first=Simon|date=20 November 1999|publisher=Laboratoire de combinatoire et d'informatique mathématique|access-date=5 May 2009}}</ref> |
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The MRB constant is defined as the [[upper limit]] of the sums<ref name="Weisstein" /><ref>{{cite arXiv| |
The MRB constant is defined as the [[upper limit]] of the [[partial sums]]<ref name="Weisstein" /><ref>{{cite arXiv|eprint=0912.3844|first=Richard J.|last=Mathar|title=Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity|year=2009|class=math.CA}}</ref><ref>{{cite web|url=http://www.perfscipress.com/papers/UniversalTOC25.pdf|title=Unified algorithms for polylogarithm, L-series, and zeta variants|last=Crandall|first=Richard|publisher=PSI Press|archive-url=https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf|archive-date=April 30, 2013|url-status=usurped|access-date=16 January 2015}}</ref><ref>{{OEIS|id=A037077}}</ref><ref>{{OEIS|id=A160755}}</ref><ref>{{OEIS|id=A173273}}</ref><ref>{{cite web|url=http://www.bitman.name/math/article/962|title=MRB (costante)|last=Fiorentini|first=Mauro|website=bitman.name|language=italian|access-date=14 January 2015}}</ref> |
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: <math>s_n = \sum_{k=1}^n (-1)^k k^{1/k}</math> |
: <math>s_n = \sum_{k=1}^n (-1)^k k^{1/k}</math> |
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As <math>n</math>grows to infinity, the sums have upper and lower |
As <math>n</math> grows to infinity, the sums have [[Limit inferior and limit superior|upper and lower limit points]] of −0.812140… and 0.187859…, separated by an [[interval (mathematics)|interval]] of length 1. The constant can also be explicitly defined by the following infinite sums:<ref name="Weisstein">{{MathWorld |title=MRB Constant |urlname=MRBConstant}}</ref> |
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: <math>0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right).</math> |
: <math>0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right).</math> |
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:<math>\sum_{k=1}^{\infty} (-1)^k k^{1/k}.</math> |
:<math>\sum_{k=1}^{\infty} (-1)^k k^{1/k}.</math> |
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There is no known [[closed-form expression]] of the MRB constant,<ref>{{cite book|title=Mathematical Constants|last=Finch|first=Steven R.|publisher=[[Cambridge University Press]]|year=2003|isbn=0-521-81805-2|location=[[Cambridge, England]]|page=450}}</ref> nor is it known whether the MRB constant is [[algebraic number|algebraic]], [[Transcendental number|transcendental]] or even [[irrational number|irrational]]. |
There is no known [[closed-form expression]] of the MRB constant,<ref>{{cite book|title=Mathematical Constants|url=https://archive.org/details/mathematicalcons0000finc|url-access=registration|last=Finch|first=Steven R.|publisher=[[Cambridge University Press]]|year=2003|isbn=0-521-81805-2|location=[[Cambridge, England]]|page=[https://archive.org/details/mathematicalcons0000finc/page/450 450]}}</ref> nor is it known whether the MRB constant is [[algebraic number|algebraic]], [[Transcendental number|transcendental]] or even [[irrational number|irrational]]. |
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==References== |
==References== |
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==External links== |
==External links== |
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{{Portal|Mathematics}} |
{{Portal|Mathematics}} |
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* [http://marvinrayburns.com/ Official site of M.R. Burns, constant's |
* [http://marvinrayburns.com/ Official site of M.R. Burns, constant's namesake and discoverer] |
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[[Category:Mathematical constants]] |
[[Category:Mathematical constants]] |
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Latest revision as of 04:31, 21 December 2024
The MRB constant is a mathematical constant, with decimal expansion 0.187859… (sequence A037077 in the OEIS). The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999.[1] Burns had initially called the constant "rc" for root constant[2] but, at Simon Plouffe's suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant".[3]
The MRB constant is defined as the upper limit of the partial sums[4][5][6][7][8][9][10]
As grows to infinity, the sums have upper and lower limit points of −0.812140… and 0.187859…, separated by an interval of length 1. The constant can also be explicitly defined by the following infinite sums:[4]
The constant relates to the divergent series:
There is no known closed-form expression of the MRB constant,[11] nor is it known whether the MRB constant is algebraic, transcendental or even irrational.
References
[edit]- ^ Plouffe, Simon. "mrburns". Retrieved 12 January 2015.
- ^ Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
- ^ Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.
- ^ a b Weisstein, Eric W. "MRB Constant". MathWorld.
- ^ Mathar, Richard J. (2009). "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844 [math.CA].
- ^ Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original on April 30, 2013. Retrieved 16 January 2015.
- ^ (sequence A037077 in the OEIS)
- ^ (sequence A160755 in the OEIS)
- ^ (sequence A173273 in the OEIS)
- ^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
- ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
