MRB constant
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]
- [math]\displaystyle{ s_n = \sum_{k=1}^n (-1)^k k^{1/k} }[/math]
As [math]\displaystyle{ n }[/math] 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]
- [math]\displaystyle{ 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]
The constant relates to the divergent series:
- [math]\displaystyle{ \sum_{k=1}^{\infty} (-1)^k k^{1/k}. }[/math]
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
- ↑ Plouffe, Simon. "mrburns". http://www.plouffe.fr/simon/constants/mrburns.txt.
- ↑ Burns, Marvin R. (23 January 1999). "RC". http://math2.org/mmb/thread/901.
- ↑ Plouffe, Simon (20 November 1999). "Tables of Constants". Laboratoire de combinatoire et d'informatique mathématique. http://www.plouffe.fr/simon/articles/Tableofconstants.pdf.
- ↑ 4.0 4.1 Weisstein, Eric W.. "MRB Constant". http://mathworld.wolfram.com/MRBConstant.html.
- ↑ 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". PSI Press. http://www.perfscipress.com/papers/UniversalTOC25.pdf.
- ↑ (sequence A037077 in the OEIS)
- ↑ (sequence A160755 in the OEIS)
- ↑ (sequence A173273 in the OEIS)
- ↑ Fiorentini, Mauro. "MRB (costante)" (in italian). http://www.bitman.name/math/article/962.
- ↑ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2. https://archive.org/details/mathematicalcons0000finc.
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