A072449 Decimal expansion of the limit of the nested radical sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

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

Offset

1,2

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 A072450.

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + …))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, … for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

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.

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Geoffrey B. Campbell and A. Zujev, Some left nested radicals, Preprint 2016.

A. Herschfeld, On Infinite Radicals, Amer. Math. Monthly, 42 (1935), 419-429.

Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.

Eric Weisstein's World of Mathematics, Nested Radical

Eric Weisstein's World of Mathematics, Nested Radical Constant

Wikipedia, Tirukkannapuram Vijayaraghavan

Example

Sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... =~ 1.757932756618004532708819638218138527653...

Mathematica

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 10, 111][[1]] (* A New Kind Of Science *)

Prog

(PARI) s=200; for(n=1, 199, t=200-n+sqrt(s); s=t); sqrt(s) \\ gives at least 180 correct digits

Crossrefs

Cf. A072450, A239349.

Sequence in context: A230163 A143297 A195348 * A263770 A088839 A111769

Adjacent sequences: A072446 A072447 A072448 * A072450 A072451 A072452

Keyword

nonn,cons

Author

Robert G. Wilson v, Aug 01 2002

Status

approved