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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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6
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| From Daniel Forgues, April 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)
Let MRB=MRB constant, then the value for a such that limit(sum((-1)^m*(m^(1/m)-a), m = 1 .. 2*N), N = infinity) = MRB and limit(sum((-1)^m*(a-m^(1/m)), m = 1 .. 2*N-1), N = infinity) = MRB is A173273. [From Marvin Ray Burns, Apr 25 2010]
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.
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LINKS
| Robert G. Wilson v, Table of n, a(n) for n = 0..1000
M. R. Burns, 300,000 digits Computed Oct 08, 2011 to Nov 5, 2011
R. J. Mathar Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity
Simon Plouffe Tables of Constants [From Laboratoire de combinatoire et d'informatique mathématique]
Eric Weisstein's World of Mathematics, Power Tower
Eric Weisstein's World of Mathematics, MRB Constant
OEIS Wiki, MRB constant
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EXAMPLE
| =0.1878596424620671202485179340542732300559030949001387861720046840894772315...
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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 }
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MATHEMATICA
| 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
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PROG
| (PARI) sumalt(x=1, (-1)^x*((x^(1/x))-1))
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CROSSREFS
| Cf. A052110, A157852, A160755.
Sequence in context: A180311 A103984 A203914 * A094106 A021536 A199389
Adjacent sequences: A037074 A037075 A037076 * A037078 A037079 A037080
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KEYWORD
| cons,nonn
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AUTHOR
| Marvin Ray Burns (bmmmburns(AT)sbcglobal.net). Entry updated Jan 30 2009, Jun 21 2009, Dec 11 2009, Sep 04 2010, June 23, 2011
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EXTENSIONS
| Definition corrected by Daniel Forgues (kephalopod(AT)gmail.com), Apr 20 2011
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