# 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

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]

The MRB constant can be explicitly defined by the following infinite sums:[2]

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

There is no known closed-form expression of the MRB constant.[3]

## History

Marvin Ray Burns published his discovery of the constant in 1999. The discovery is a result of a "math binge" that started in the spring of 1994.[4] 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.[5] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.[6]

## References

1. ^ Burns, Marvin R. (1999-09-01). "MRB Constant". Plouffe's Inverter. Retrieved 2009-05-05.
2. ^
3. ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
4. ^ Burns, Marvin R. (2002-04-12). "Captivity’s Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 2009-05-05.
5. ^ Burns, Marvin R. (1999-01-23). "RC". math2.org. Retrieved 2009-05-05.
6. ^ Plouffe, Simon (1999-11-20). "Tables of Constants". Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05.