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A037077
Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... .
8
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, 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, 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
OFFSET
0,2
COMMENTS
From Daniel Forgues, Apr 20 2011: (Start)
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.
The series Sum_{n>=1} (-1)^n (n^(1/n)-1) converges to this upper limit. (End)
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
LINKS
Marvin Ray Burns, Mathematica Notebook of first known 314159 digit computation, finished on Sep 04 2012.
Marvin Ray Burns, Mathematica Notebook of first known 3014991 digit computation, finished on Sep 21 2014.
OEIS Wiki, MRB constant
Simon Plouffe, From tables of Constants [Original documentation written by M. R. Burns in 1999]
Eric Weisstein's World of Mathematics, MRB Constant
Eric Weisstein's World of Mathematics, Power Tower
EXAMPLE
0.1878596424620671202485179340542732300559030949001387861720046840894772315...
MAPLE
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;
A037077(1000) # { where 1000 is the number of digits desired }
MATHEMATICA
(* Program 1 *)
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]];
RealDigits[ f[105], 10][[1]] (* mx is the number of digits desired - Marvin Ray Burns, Aug 05 2007 *)
(* Program 2 *)
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 *)
(* Program 3 *)
(* Fastest as of Jan 06 2013. For use with large calculations (5, 000-3, 000, 000 digits) *)
prec = 5000; (* Number of required digits. *)
ClearSystemCache[];
T0 = SessionTime[];
expM[pre_] :=
Module[{a, d, s, k, bb, c, n, end, iprec, xvals, x, pc, cores = 6,
tsize = 2^7, chunksize, start = 1, ll, ctab,
pr = Floor[1.02 pre]}, chunksize = cores*tsize;
n = Floor[1.32 pr];
end = Ceiling[n/chunksize];
Print["Iterations required: ", n];
Print["end ", end];
Print[end*chunksize]; d = ChebyshevT[n, 3];
{b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};
iprec = Ceiling[pr/27];
Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;
x = N[E^(Log[ll]/(ll)), iprec];
pc = iprec;
While[pc < pr, pc = Min[3 pc, pr];
x = SetPrecision[x, pc];
y = x^ll - ll;
x = x (1 - 2 y/((ll + 1) y + 2 ll ll)); ]; (*N[Exp[Log[ll]/ll],
pr]*)x, {l, 0, tsize - 1}], {j, 0, cores - 1},
Method -> "CoarsestGrained"]];
ctab = ParallelTable[Table[c = b - c;
ll = start + l - 2;
b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));
c, {l, chunksize}], Method -> "CoarsestGrained"];
s += ctab.(xvals - 1);
start += chunksize;
Print["done iter ", k*chunksize, " ", SessionTime[] - T0]; , {k, 0,
end - 1}];
N[-s/d, pr]];
t2 = Timing[MRBtest2 = expM[prec]; ];
Print[MRBtest2] (* Richard Crandall via Marvin Ray Burns, Feb 19 2013 *)
PROG
(PARI) sumalt(x=1, (-1)^x*((x^(1/x))-1))
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Marvin Ray Burns; entry updated Jan 30 2009, Jun 21 2009, Dec 11 2009, Sep 04 2010, Jun 23 2011, Sep 08 2012
EXTENSIONS
Definition corrected by Daniel Forgues, Apr 20 2011
STATUS
approved