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A160755
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Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}]
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1
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1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| If one would fail to use acceleration methods, then according to this sequence 10^49 terms must be added to arrive at 50 digits of the MRB constant.
So at 10^10 Hertz and with 2^16 terms processed per Hertz, it would take 176,606,354,890,046,296,296,296,296,296 days to compute 50 digits of the MRB constant.
Compute days as follows:
The 50th term is 49
and
rate = 10^10*2^16
days = 10^49/rate/3600/24.
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REFERENCES
| Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, "Convergence Acceleration of Alternating Series", Experimental Mathematics, 9:1 (2000).
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.
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LINKS
| Anonymous, Table of selected mathematical constants, Wikipedia.
Eric W. Weisstein, MRB Constant, Mathworld.
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FORMULA
| Roughly equal to A004709
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EXAMPLE
| After 10^1 partial sums you get one accurate digit; 10^2 partial sums = two accurate digits and so on.
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MATHEMATICA
| m = NSum[(-1)^n*(n^(1/n) - 1), {n, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 1000]; Table[-Floor[Log[10, Abs[m - NSum[(-1)^n*(n^(1/n) - 1), {n, 10^a}, Method ->"AlternatingSigns", WorkingPrecision -> 1000]]]], {a, 1, 50}]
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CROSSREFS
| Cf. A037077 (the MRB constant).
Sequence in context: A108922 A102670 A079631 * A017873 A128557 A103303
Adjacent sequences: A160752 A160753 A160754 * A160756 A160757 A160758
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KEYWORD
| nonn,base,less
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AUTHOR
| Marvin Ray Burns (bmmmburns(AT)sbcglobal.net), May 25 2009
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EXTENSIONS
| Corrections from Marvin Ray Burns (bmmmburns(AT)sbcglobal.net), Jun 05 2009
Link to wikipedia replaced by up-to-date version; keyword:less added R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 04 2010
Correction by Marvin Ray Burns (bmmmburns(AT)sbcglobal.net), Aug 21 2010
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