MRB constant explained

The MRB constant is a mathematical constant, with decimal expansion . 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]

s_n=

	n
\sum
	k=1

(-1)^kk^1/k

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:

0.187859\ldots=

	infty
\sum
	k=1

(-1)^k(k^1/k-1)=

	infty
\sum
	k=1

\left((2k)^1/(2k)-(2k-1)^1/(2k-1)\right).

The constant relates to the divergent series:

	infty
\sum
	k=1

(-1)^kk^1/k.

There is no known closed-form expression of the MRB constant,^[7] nor is it known whether the MRB constant is algebraic, transcendental or even irrational.

External links

Official site of M.R. Burns, constant's namesake and discoverer

Notes and References

Web site: mrburns. Plouffe. Simon. 12 January 2015.
Web site: RC. Burns. Marvin R.. 23 January 1999. math2.org. 5 May 2009.
Web site: Tables of Constants. Plouffe. Simon. 20 November 1999. Laboratoire de combinatoire et d'informatique mathématique. 5 May 2009.
0912.3844. Richard J.. Mathar. Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity. 2009. math.CA.
Web site: Unified algorithms for polylogarithm, L-series, and zeta variants. Crandall. Richard. PSI Press. https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf. April 30, 2013. dead. 16 January 2015.
Web site: MRB (costante). Fiorentini. Mauro. bitman.name. italian. 14 January 2015.
Book: Finch, Steven R.. Mathematical Constants. registration. Cambridge University Press. 2003. 0-521-81805-2. Cambridge, England. 450.