

A072449


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


29



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
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OFFSET

1,2


COMMENTS

No closedform 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.
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

Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC353 to PC354. Annotated and scanned copy.


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=200n+sqrt(s); s=t); sqrt(s) \\ gives at least 180 correct digits


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



