Consider the sequence of partial sums defined by

(1)

As can be seen in the plot above, the sequence has two limit points at 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
as

where

(9)

and denotes the th derivative of evaluated at (Crandall 2012ab).

An integral expression for the constant is given by

(10)

(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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References Burns, M. R. "An Alternating Series Involving Roots." Unpublished note,
1999. Burns, M. R. "Try to Beat These MRB Constant Records!"
http://community.wolfram.com/groups/-/m/t/366628 . Crandall,
R. E. "Unified Algorithms for Polylogarithm, -Series, and Zeta Variants." 2012a. http://www.marvinrayburns.com/UniversalTOC25.pdf . Crandall,
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." http://pi.lacim.uqam.ca/piDATA/mrburns.txt . Sloane,
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. https://mathworld.wolfram.com/MRBConstant.html

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