MRB constant

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]

First 100 partial sums of ${\displaystyle (-1)^{k}(k^{1/k}-1)}$

The MRB constant is defined as the upper limit of the partial sums^{[4][5][6][7][8]}

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

As ${\displaystyle n}$ 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]

${\displaystyle 0.187859\ldots =\sum _{k=1}^{\infty }(-1)^{k}(k^{1/k}-1)=\sum _{k=1}^{\infty }\left((2k)^{1/(2k)}-(2k-1)^{1/(2k-1)}\right).}$

The constant relates to the divergent series:

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

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.