MRB constant

Mathematical constant described by Marvin Ray Burns
First 100 partial sums of ( − 1 ) k ( k 1 / k − 1 ) {\displaystyle (-1)^{k}(k^{1/k}-1)}

The MRB constant is a mathematical constant, with decimal expansion 0.187859… (sequence A037077 in the OEIS). 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][7][8][9][10]

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

As n {\displaystyle n} 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:[4]

0.187859 … = ∑ k = 1 ∞ ( − 1 ) k ( k 1 / k − 1 ) = ∑ k = 1 ∞ ( ( 2 k ) 1 / ( 2 k ) − ( 2 k − 1 ) 1 / ( 2 k − 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).}

The constant relates to the divergent series:

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

There is no known closed-form expression of the MRB constant,[11] nor is it known whether the MRB constant is algebraic, transcendental or even irrational.

References

  1. ^ Plouffe, Simon. "mrburns". Retrieved 12 January 2015.
  2. ^ Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
  3. ^ Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.
  4. ^ a b Weisstein, Eric W. "MRB Constant". MathWorld.
  5. ^ Mathar, Richard J. (2009). "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844 [math.CA].
  6. ^ 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.
  7. ^ (sequence A037077 in the OEIS)
  8. ^ (sequence A160755 in the OEIS)
  9. ^ (sequence A173273 in the OEIS)
  10. ^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
  11. ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.

External links

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