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A037077 Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... .

%I A037077 #153 Feb 16 2025 08:32:37
%S A037077 1,8,7,8,5,9,6,4,2,4,6,2,0,6,7,1,2,0,2,4,8,5,1,7,9,3,4,0,5,4,2,7,3,2,
%T A037077 3,0,0,5,5,9,0,3,0,9,4,9,0,0,1,3,8,7,8,6,1,7,2,0,0,4,6,8,4,0,8,9,4,7,
%U A037077 7,2,3,1,5,6,4,6,6,0,2,1,3,7,0,3,2,9,6,6,5,4,4,3,3,1,0,7,4,9,6,9,0,3,8,4,2
%N A037077 Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... .
%C A037077 From _Daniel Forgues_, Apr 20 2011: (Start)
%C A037077 The series Sum_{n>=1} (-1)^n n^(1/n) diverges (oscillates) with the upper limit given by this sequence and the lower limit being the upper limit - 1.
%C A037077 The series Sum_{n>=1} (-1)^n (n^(1/n)-1) converges to this upper limit. (End)
%D A037077 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
%H A037077 Robert G. Wilson v, <a href="/A037077/b037077.txt">Table of n, a(n) for n = 0..1000</a>
%H A037077 Marvin Ray Burns, <a href="/A037077/a037077.nb">Mathematica Notebook of first known 314159 digit computation</a>, finished on Sep 04 2012.
%H A037077 Marvin Ray Burns, <a href="http://marvinrayburns.com/314159.txt">Text version of 314159 digits</a>
%H A037077 Marvin Ray Burns, <a href="/A037077/a037077_1.nb">Mathematica Notebook of first known 3014991 digit computation</a>, finished on Sep 21 2014.
%H A037077 Marvin Ray Burns, <a href="http://www.marvinrayburns.com/6029991A037077.nb">Mathematica Notebook of first known 6029991 digit computation of A037077</a>, finished on Mar 30 2021.
%H A037077 Richard E. Crandall, <a href="http://www.marvinrayburns.com/UniversalTOC25.pdf">Unified algorithms for polylogarithm, L-series, and zeta variants</a> (53 pages).
%H A037077 OEIS Wiki, <a href="/wiki/MRB_constant">MRB constant</a>
%H A037077 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/mrburns.txt"> From tables of Constants</a> [Original documentation written by M. R. Burns in 1999]
%H A037077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MRBConstant.html">MRB Constant</a>
%H A037077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%e A037077 0.1878596424620671202485179340542732300559030949001387861720046840894772315...
%p A037077 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;
%p A037077 A037077(1000)  # { where 1000 is the number of digits desired }
%t A037077 (* Program 1 *)
%t A037077 f[mx_] := Block[{$MaxExtraPrecision = mx + 8, a, b = -1, c = -1 - d, d = (3 + Sqrt[8])^n, n = 131 Ceiling[mx/100], s = 0}, a[0] = 1; d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++]; For[k = 0, k < n, c = b - c; b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; N[1/2 - s/d, mx]];
%t A037077 RealDigits[ f[105], 10][[1]] (* mx is the number of digits desired  - _Marvin Ray Burns_, Aug 05 2007 *)
%t A037077 (* Program 2 *)
%t A037077 digits = 105; NSum[ (-1)^n*((n^(1/n)) - 1), {n, 1, Infinity}, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 15 2013 *)
%t A037077 (* Program 3 *)
%t A037077 (* Fastest as of Jan 06 2013. For use with large calculations (5,000-3,000,000 digits) *)
%t A037077 prec = 5000; (* Number of required digits. *)
%t A037077 ClearSystemCache[];
%t A037077 T0 = SessionTime[];
%t A037077 expM[pre_] :=
%t A037077   Module[{a, d, s, k, bb, c, n, end, iprec, xvals, x, pc, cores = 6,
%t A037077     tsize = 2^7, chunksize, start = 1, ll, ctab,
%t A037077     pr = Floor[1.02 pre]}, chunksize = cores*tsize;
%t A037077    n = Floor[1.32 pr];
%t A037077    end = Ceiling[n/chunksize];
%t A037077    Print["Iterations required: ", n];
%t A037077    Print["end ", end];
%t A037077    Print[end*chunksize]; d = ChebyshevT[n, 3];
%t A037077    {b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};
%t A037077    iprec = Ceiling[pr/27];
%t A037077    Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;
%t A037077         x = N[E^(Log[ll]/(ll)), iprec];
%t A037077         pc = iprec;
%t A037077         While[pc < pr, pc = Min[3 pc, pr];
%t A037077          x = SetPrecision[x, pc];
%t A037077          y = x^ll - ll;
%t A037077          x = x (1 - 2 y/((ll + 1) y + 2 ll ll));];(*N[Exp[Log[ll]/ll],
%t A037077         pr]*)x, {l, 0, tsize - 1}], {j, 0, cores - 1},
%t A037077        Method -> "CoarsestGrained"]];
%t A037077     ctab = ParallelTable[Table[c = b - c;
%t A037077        ll = start + l - 2;
%t A037077        b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));
%t A037077        c, {l, chunksize}], Method -> "CoarsestGrained"];
%t A037077     s += ctab.(xvals - 1);
%t A037077     start += chunksize;
%t A037077     Print["done iter ", k*chunksize, " ", SessionTime[] - T0];, {k, 0,
%t A037077       end - 1}];
%t A037077    N[-s/d, pr]];
%t A037077 t2 = Timing[MRBtest2 = expM[prec];];
%t A037077 Print[MRBtest2] (* Richard Crandall via _Marvin Ray Burns_, Feb 19 2013 *)
%o A037077 (PARI) sumalt(x=1,(-1)^x*((x^(1/x))-1))
%Y A037077 Cf. A052110, A157852, A160755, A173273.
%K A037077 cons,nonn
%O A037077 0,2
%A A037077 _Marvin Ray Burns_; entry updated Jan 30 2009, Jun 21 2009, Dec 11 2009, Sep 04 2010, Jun 23 2011, Sep 08 2012
%E A037077 Definition corrected by _Daniel Forgues_, Apr 20 2011