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. Retrieved 16 January 2015.
  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.
  13. ↑ Plouffe, Simon (1999-11-20). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05.

External links

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