The MRB constant can be computed in Maple by evalf(sum((-1)^n*(n^(1/n)-1),n=1..infinity)).

On my laptop restart; st := time(); evalf(sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity), 500); time()-st gives a timming of 37.908 seconds.

Using the procedure posted at the bottom of this message st := time(); A037077(500); time()-st gives a much faster timing of 1.903 seconds.

My fastest timing for 500 digits of MRB comes from my desktop with Mathematica 6.02 using the command 

NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]}, WorkingPrecision -> 500, 
  Method -> "AlternatingSigns"] // Timing

giving a timing of 0.187 seconds.

I was wondering if there are still faster ways to compute the MRB constant in Maple. Also I would like to have a professional looking Maple Cloud document showing various ways of computing the constant in Maple.

Richard E. Crandall published an unified method of computing the MRB constant along with other constants at http://www.perfscipress.com/papers/UniversalTOC25.pdf . I wonder if it can help?

The procedure I used could stand some improvement and might benefit from parallelization:

 A037077 := proc (e) local a, b, c, d, s, k, n, m; if e < 100 then n := 31+e; Digits := 31+e else n := 131*round((1/100)*e); Digits := 131*round((1/100)*e) end if; a := array(0 .. n-1); a[0] := 1; for m to n-1 do a[m] := ((1/2)*sinh(2*ln(m+1)/(m+1))+cosh(ln(m+1)/(m+1))^2-1)/sinh(ln(m+1)/(m+1)) end do; d := (1/2)*(3+2*2^(1/2))^n+(1/2)/(3+2*2^(1/2))^n; b := -1; c := -d; s := 0; for k from 0 to n-1 do c := b-c; b := 2*b*(k^2-n^2)/((2*k+1)*(k+1)); s := s+c*a[k] end do; Digits := e; print(evalf(1/2-s/d)) end proc