C_MRB is defined below. See http://mathworld.wolfram.com/MRBConstant.html.

Starting by using Maple on the Inverse Symbolic Calculator, with over 21 years of research and ideas from users like you, I developed this shortlist of formulas for the MRB constant.

• C_MRB=  

That is proven below by an internet scholar going by the moniker "Dark Malthorp:"

•  denoting the kth derivative of the Dirichlet eta function of k and 0 respectively was first discovered in 2012 by Richard Crandall of Apple Computer.

The left half is proven below by Gottfried Helms and it is proven more rigorously considering the conditionally convergent sum, below that. Then the right half is a Taylor expansion of η(s) around s = 0.

At https://math.stackexchange.com/questions/1673886/is-there-a-more-rigorous-way-to-show-these-two-sums-are-exactly-equal,

it has been noted that "even though one has cause to be a little bit wary around formal rearrangements of conditionally convergent sums (see the Riemann series theorem), it's not very difficult to validate the formal manipulation of Helms. The idea is to cordon off a big chunk of the infinite double summation (all the terms from the second column on) that we know is absolutely convergent, which we are then free to rearrange with impunity. (Most relevantly for our purposes here, see pages 80-85 of this document, culminating with the Fubini theorem which is essentially the manipulation Helms is using.)"