The MRB constant ("C"MRB) =

Here are three programs for calculating digits of "C"MRB.

Can you prove why they work?

Here is the first one:

1/2 - Pi/2 + 
 NSum[(-1)^n*(n^(1/n) - Pi), {n, 1, \[Infinity]}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 3000, 
  PrecisionGoal -> 3000]

Here is the second one:

eta[s_] := (1 - 2^(1 - s)) Zeta[s];
a[i_] := Derivative[i][eta][0];
c[j_] := Sum[Binomial[j, d]*(-1)^d*d^(j - d), {d, 1, j}];
-N[Sum[c[m]/m!*a[m], {m, 1, 40}], 6]

Here is one that is not so short:

(*Newer loop with Newton interior.*)prec = 1000;(*Number of required \
decimals.*)expM[pr_] := Module[{a, d, s, k, b, c}, n = Floor[1.32 pr];
  Print["Iterations required: ", n];
  d = N[(3 + Sqrt[8])^n, pr + 10];
  d = Round[1/2 (d + 1/d)];
  {b, c, s} = {-1, -d, 0};
  T0 = SessionTime[];
  Do[c = b - c;
   x = N[E^(Log[k + 1]/(k + 1)), iprec = Ceiling[prec/128]];
   pc = iprec;
   Do[nprec = Min[2 pc, pr];
    x = SetPrecision[x, nprec];(*Adjust precision*)
    x = N[x - x/(k + 1) + 1/x^k, nprec];
    pc *= 2;
    If[nprec >= pr, Break[]], {ct, 1, 19}];
   s += c*(x - 1);
   b *= 2 (k + n) (k - n)/((k + 1) (2 k + 1));
   If[Mod[k, 1000] == 0, 
    Print["Iterations: ", k, "    Cumulative time (sec): ", 
      SessionTime[] - T0];], {k, 0, n - 1}];
  N[-s/d, pr]];

t2 = Timing[MRBtest2 = expM[prec];]

If you can come up with others, please post them!

I think the last two are related, somehow.

Here is some help on proving why the first one works: (See attached notebook.)

Attachments: