Henrik et al,

I have a big job ahead of me if I try to use Shanks to break my record of calculated digits of the MRB constant!! The terms needed to break my record of 3,014,991 digits seems impractical to list! Maybe I just need a new way of computing the Shanks' terms (someway of using known acceleration methods to accumulate them).

Thank you Henrik! I think the Shanks transformation will help me. It seems, however, Mathematica has an issue with 10's and 100's of iterations of Shanks.

Again. let g[n] be the error in using f, as you use 10^n terms, as shown in the original post; using the following piece of code, and then it appears to be the case that the g[10 n]'s are rational factors of Log[10]^3, to better and better precision as n gets large:

 m = 
  NSum[(-1)^k (k^(1/k) - 1), {k, 1, Infinity}, 
   WorkingPrecision -> 1000, Method -> "AlternatingSigns"];



 f[k_] := Log[(k - Log[k])/k];

ClearAll[g];
g[n_] := Module[{}, 
  m - (NSum[(-1)^k (k^(1/k) - 1 + f[k]), {k, 1, 10^n}, 
      WorkingPrecision -> Floor[3 n + 100], 
      Method -> "AlternatingSigns"] - 
     NSum[(-1)^k*f[k], {k, 1, Infinity}, 
      WorkingPrecision -> Floor[3 n + 100], 
      Method -> "AlternatingSigns"])]



 Table[
 RootApproximant[MantissaExponent[Log[10]^3/g[10 n]][[1]]], {n, 2, 20}]

(* {3/20, 4/9, 3/16, 24/25, 5/9, 120/343, 15/64, 40/243, 3/25, \ 1200/1331, 25/36, 1200/2197, 150/343, 16/45, 75/256, 1200/4913, \ 50/243, 1200/6859, 3/20} *)