# 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 the OEIS).

## Definition

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

The MRB constant is related to the following divergent series:

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

Its partial sums

${\displaystyle s_{n}=\sum _{k=1}^{n}(-1)^{k}k^{1/k}}$

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:[1]

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

There is no known closed-form expression of the MRB constant.[8]

## History

Marvin Ray Burns published his discovery of the constant in 1999.[9] The discovery is a result of a "math binge" that started in the spring of 1994.[10] 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.[11] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.[12]

## References

1. ^ a b Weisstein, Eric W. "MRB Constant". MathWorld.
2. ^ Mathar, Richard J. "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844.
3. ^ Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original (PDF) on April 30, 2013. Retrieved 16 January 2015.
4. ^ (sequence A037077 in the OEIS)
5. ^ (sequence A160755 in the OEIS)
6. ^ (sequence A173273 in the OEIS)
7. ^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
8. ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
9. ^ Burns, Marvin. "mrburns". Simeon Plouffe. Retrieved 12 January 2015.
10. ^ Burns, Marvin R. (12 April 2002). "Captivity's Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 5 May 2009.
11. ^ Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
12. ^ Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.