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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} (-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

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.

Marvin Ray Burns, Mathematica Notebook of first known 6029991 digit computation of A037077, finished on Mar 30 2021.

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-Fran├žois 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

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 May 25 09:45 EDT 2022. Contains 354066 sequences. (Running on oeis4.)