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
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  | ![sum_(k=1)^(infty)[(2k)^(1/(2k))-(2k-1)^(1/(2k-1))]](/images/equations/MRBConstant/Inline10.svg) |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(Finch 2003, p. 450; OEIS A037077).
The constant can also be given as a sum over derivatives of the Dirichlet eta function 
as
 |  |  |
(7)
|
 |  |  |
(8)
|
where
 |
(9)
|
and 
denotes the 
th derivative of 
evaluated at 
(Crandall 2012ab).
An integral expression for the constant is given by
![S=int_0^inftycsch(pit)I[(1+it)^(1/(1+it))]dt](/images/equations/MRBConstant/NumberedEquation3.svg) |
(10)
|
(M. Burns, pers. comm., Jan. 21, 2020).
No closed-form expression is known for this constant (Finch 2003, p. 450).
Roots." Unpublished note,
1999.Burns, M. R. "Try to Beat These MRB Constant Records!"
-Series, and Zeta Variants." 2012a.