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A037077
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Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... .
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8
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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
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OFFSET
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0,2
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COMMENTS
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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)
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
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LINKS
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EXAMPLE
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0.1878596424620671202485179340542732300559030949001387861720046840894772315...
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MAPLE
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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 }
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MATHEMATICA
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(* 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]; ];
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PROG
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(PARI) sumalt(x=1, (-1)^x*((x^(1/x))-1))
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CROSSREFS
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KEYWORD
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AUTHOR
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Marvin Ray Burns; entry updated Jan 30 2009, Jun 21 2009, Dec 11 2009, Sep 04 2010, Jun 23 2011, Sep 08 2012
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EXTENSIONS
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STATUS
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approved
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