MRB constant

{{Short description|Mathematical constant described by Marvin Ray Burns}}

File:MRB-Gif.gifThe 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.{{cite web|url=http://www.plouffe.fr/simon/constants/mrburns.txt|title=mrburns|last=Plouffe|first=Simon|access-date=12 January 2015}} Burns had initially called the constant "rc" for root constant{{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}} but, at Simon Plouffe's suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant".{{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}}

The MRB constant is defined as the upper limit of the partial sums{{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}}{{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}}{{OEIS|id=A037077}} {{OEIS|id=A160755}} {{OEIS|id=A173273}}{{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}}

: $s\_n\; =\; \backslash sum\_\{k=1\}^n\; (-1)^k\; k^\{1/k\}$

As $n$ 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:{{MathWorld |title=MRB Constant |urlname=MRBConstant}}

: $0.187859\backslash ldots\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; (-1)^k\; (k^\{1/k\}\; -\; 1)\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash left((2k)^\{1/(2k)\}\; -\; (2k-1)^\{1/(2k-1)\}\backslash right).$

The constant relates to the divergent series:

:$\backslash sum\_\{k=1\}^\{\backslash infty\}\; (-1)^k\; k^\{1/k\}.$

There is no known closed-form expression of the MRB constant,{{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]}} nor is it known whether the MRB constant is algebraic, transcendental or even irrational.