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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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Daniel Forgues, Apr 20 2011: (Start) The series Sum_(n=1..infty) (-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..infty) (-1)^n (n^(1/n)-1) converges to this upper limit. (End) This sequence is also the decimal expansion of the MRB constant. - Marvin Ray Burns, Jan 13 2015 Added a program that Richard Crandall wrote (and I tweaked a little). See the link below, sections 7.5 and 7.6. - Marvin Ray Burns Feb 19 2013 3014991 terms of this sequence are known. - Marvin Ray Burns, Sep 27 2014 REFERENCES S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452. LINKS Robert G. Wilson v, Table of n, a(n) for n = 0..1000 Marvin Ray Burns, Mathematica Notebook of first known 314159 digit computation finished on Sep 04 2012 Marvin Ray Burns, Text version of 314159 digits Marvin Ray Burns, Mathematica Notebook of first known 3014991 digit computation finished on Sep 21 2014 Richard E. Crandall, Unified algorithms for polylogarithm, L-series, and zeta variants (53 pages) 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 (at MRB's end) as of Jan 06 2013. For use with large calculations (5, 000-3, 000, 000 digits) *) prec = 5000; (* Number of required decimals. *) 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 Cf. A052110, A157852, A160755, A173273. Sequence in context: A289913 A103984 A203914 * A094106 A277064 A276762 Adjacent sequences:  A037074 A037075 A037076 * A037078 A037079 A037080 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

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Last modified December 12 16:06 EST 2018. Contains 318077 sequences. (Running on oeis4.)