MRB constant

Marvin R. Burns, the constant's author, in 1999

The MRB constant, named after Marvin Ray Burns, is a mathematical constant for which no closed-form expression is known. It is not known whether the MRB constant is algebraic, transcendental, or even irrational.

The numerical value of MRB constant, truncated to 6 decimal places, is

0.187859… (sequence A037077 in OEIS).

Definition

First 100 partial sums of $(-1)^k (k^{1/k} - 1)$

The MRB constant is related to the following divergent series:

$\sum_{k=1}^{\infty} (-1)^k k^{1/k}.$

Its partial sums

$s_n = \sum_{k=1}^n (-1)^k k^{1/k}$

are bounded so that their limit points form an interval [−0.812140…,0.187859…] of length 1. The upper limit point 0.187859… is what is known as the MRB constant.^{[1][2][3][4][5][6][7]}

The MRB constant can be explicitly defined by the following infinite sums:^[8]

$0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right).$

There is no known closed-form expression of the MRB constant.^[9]

History

History Marvin Ray Burns published his discovery of the constant in 1999.^[10] The discovery is a result of a "math binge" that started in the spring of 1994.^[11] Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant.^[12] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.^[13]

References

1. Weisstein, Eric W. ""MRB Constant.". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 12 January 2015.
2. MATHAR, RICHARD J. "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^*1/x) BETWEEN 1 AND INFINITY" (PDF). arxiv. Cornell University. Retrieved 12 January 2015.
3. Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). http://web.archive.org/. PSI Press. Archived from the original (PDF) on April 30, 2013. Retrieved 16 January 2015. External link in |website= (help)
4. (sequence A037077 in OEIS)
5. (sequence A160755 in OEIS)
6. (sequence A173273 in OEIS)
7. Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Mauro Fiorentini. Retrieved 14 January 2015.
8. Weisstein, Eric W., "MRB Constant", MathWorld.
9. Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
10. Burns, Marvin. "mrburns.". plouffe.fr. SImeon Plouffe. Retrieved 12 January 2015.
11. Burns, Marvin R. (2002-04-12). "Captivity’s Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 2009-05-05.
12. Burns, Marvin R. (1999-01-23). "RC". math2.org. Retrieved 2009-05-05. External link in |publisher= (help)
13. Plouffe, Simon (1999-11-20). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05. External link in |publisher= (help)

External links

• Official site of M.R. Burns, constant's author