MRB constant

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First 100 partial sums of ( − 1 ) k ( k 1 / k − 1 ) {\displaystyle (-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]

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

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

There is no known closed-form expression of the MRB constant,[9] 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. 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 on April 30, 2013. Retrieved 16 January 2015.
  7. (sequence A037077 in the OEIS) (sequence A160755 in the OEIS) (sequence A173273 in the OEIS)
  8. Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
  9. Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.