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A072449
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Decimal expansion of the limit of the nested radical sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).
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13
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1, 7, 5, 7, 9, 3, 2, 7, 5, 6, 6, 1, 8, 0, 0, 4, 5, 3, 2, 7, 0, 8, 8, 1, 9, 6, 3, 8, 2, 1, 8, 1, 3, 8, 5, 2, 7, 6, 5, 3, 1, 9, 9, 9, 2, 2, 1, 4, 6, 8, 3, 7, 7, 0, 4, 3, 1, 0, 1, 3, 5, 5, 0, 0, 3, 8, 5, 1, 1, 0, 2, 3, 2, 6, 7, 4, 4, 4, 6, 7, 5, 7, 5, 7, 2, 3, 4, 4, 5, 5, 4, 0, 0, 0, 2, 5, 9, 4, 5, 2, 9, 7, 0, 9, 3
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Herschfeld calls this the Kasner number, after Edward Kasner. - Charles R Greathouse IV, Dec 30 2008
No closed-form expression is known for this constant.
"It was discovered by T. Vijayaraghavan that the infinite radical sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ..., where a_n >= 0, will converge to a limit if and only if the limit of (log a_n)/2^n exists" - Clawson, p. 229. Obviously if a_n = n, the limit of (log a_n) / 2^n as n -> infinity is 0.
The continued fraction is 1, 1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...
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REFERENCES
| Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.1.
Aaron Herschfeld, "On Infinite Radicals", American Mathematical Monthly 42:7 (1935), pp. 419-429.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 30.
Stephen Wolfram, "A New Kind Of Science," Wolfram Media, 2002, page 915.
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LINKS
| Eric Weisstein's World of Mathematics, Nested Radical
Eric Weisstein's World of Mathematics, Nested Radical Constant
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EXAMPLE
| Sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... =~ 1.757932756618004532708819638218138527653...
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MATHEMATICA
| RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 10, 111][[1]] A New Kind Of Science
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PROG
| (PARI) (gives at least 180 correct digits) s=200; for(n=1, 199, t=200-n+sqrt(s); s=t); sqrt(s)
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CROSSREFS
| Cf. A072450.
Sequence in context: A194657 A143297 A195348 * A088839 A111769 A111513
Adjacent sequences: A072446 A072447 A072448 * A072450 A072451 A072452
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KEYWORD
| nonn,cons
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 01 2002
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