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

Author

Robert G. Wilson v, Aug 01 2002

Keywords

• nonn
• cons

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

Examples

			1.757932756618004532708819638218138527653...
		

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.

Crossrefs

Cf. A072450, A239349.

Programs

Mathematica

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

PARI

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