MRB Constant


Consider the sequence of partial sums defined by


As can be seen in the plot above, the sequence has two limit points at -0.812140... and 0.187859... (which are separated by exactly 1). The upper limit point is sometimes known as the MRB constant after the initials of its original investigator (Burns 1999; Plouffe).

Sums for the MRB constant are given by


(Finch 2003, p. 450; OEIS A037077).

The constant can also be given as a sum over derivatives of the Dirichlet eta function eta(x) as



 c_k=sum_(i=1)^k(-1)^i(k; i)i^(k-i)

and eta^(k)(x)) denotes the kth derivative of eta(x) evaluated at x (Crandall 2012ab).

An integral expression for the constant is given by


(M. Burns, pers. comm., Jan. 21, 2020).

No closed-form expression is known for this constant (Finch 2003, p. 450).

See also

Glaisher-Kinkelin Constant, Power Tower, Steiner's Problem

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Burns, M. R. "An Alternating Series Involving n^(th) Roots." Unpublished note, 1999.Burns, M. R. "Try to Beat These MRB Constant Records!", R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012a., R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012b.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 450, 2003.Plouffe, S. "MRB Constant.", N. J. A. Sequences A037077 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

MRB Constant

Cite this as:

Weisstein, Eric W. "MRB Constant." From MathWorld--A Wolfram Web Resource.

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