0014: Article 4 (Golden Ratio and Nested Radicals)
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Have You Seen This Number N = 1.8168834242447…?
By Tito Piezas III
Abstract: “We consider the asymptotic behavior of the infinite nested radical defining the golden ratio φ and plastic constant P,
and give a conjecture on the general case.”
I. Introduction
While trying to find more commonalities between the golden ratio φ and plastic constant (see article, A Tale of Four Constants), this author came across the Paris constant. In 1987, R. Paris proved that the nested radical expression for φ given above approaches φ at a constant rate. For example, defining φn as using n = {5, 6, 7} terms respectively, then,
(1/2)(φ-φ5)(2φ)5 = 1.0977…
(1/2)(φ-φ6)(2φ)6 = 1.0983…
(1/2)(φ-φ7)(2φ)7 = 1.0985…
which is approaching the Paris constant R = 1.09864196… It seems this result can be generalized to radicals of kth power, hence includes the plastic constant and other values.
Conjecture 1: “Given the infinite nested radical,
for integer k > 1, non-zero a, and the equation (eq.1),
Let x be the root of eq.1 such that x = xn as n → ∞. Define,
Then,
for some constant C(a,k).”
II. Particular cases
For quadratic radicals k = 2, since y = a/x + 1 = x, then this is simply,
(x-xn)(2x)n → C(a,2)
which apparently is relatively easy to prove. The special case a = 1,
yields C(1,2) = 2R = 2.19728392…, with the Paris constant R = 1.09864196… (R. Paris, An Asymptotic Approximation Connected with the Golden Number, Amer. Math. Monthly, 1987). For a = 2,
as Denis Feldmann points out, the constant C(a,k) has a nice closed-form expression as,
C(2,2) = (π/2)2 = 2.46740110…
For cubic radicals, say, {a,k} = {1,3}, we get the plastic constant ,
where P = 1.32472…, the real root of x3 = x+1. Since y = 1/P+1 = 1.75488…, then for small n,
(P-P5)(3y)5 = 1.81673…
(P-P6)(3y)6 = 1.81685…
(P-P7)(3y)7 = 1.81687…
As n increases, one can see this is approaching the constant,
C(1,3) = 1.8168834242447…
mentioned in the title and accurate to 12 decimal places. And so on for other values {a,k}. Note, however, that for the special case a = 1,
C(1,2) = 2.19728…
C(1,3) = 1.81688…
C(1,4) = 1.68029…
.
.
.
C(1,152) = 1.39246…
C(1,153) = 1.39242…
C(1,154) = 1.39238…
Conjecture 2: “The sequence of constants C(a,k) for a = 1 as k → ∞ is also approaching a constant.”
Question 1: Anybody knows how to prove/disprove Conjectures 1 and 2? If indeed Conjecture 2 is true, what is this constant, or, at least, what are its first dozen or so decimal digits?
(Update, 10/4/10): Courtesy of Adam Strzebonski and Daniel Lichtblau of Wolfram Research, I managed to find C(1,k) for very high k. The convergence is VERY slow, taking powers of 10 to barely move one decimal digit:
C(1,1014) = 1.3862943611198999…
C(1,1015) = 1.3862943611198915…
C(1,1016) = 1.3862943611198907…
Using Plouffe’s Inverse Symbolic Calculator, one finds that,
2 ln(2) = 1.3862943611198906…
so it is a reasonable conjecture that the sequence of constants C(1,k) as k → ∞ has its limit 2 ln(2). Note that the expression 2 ln(2) also appears in Lévy’s constant γ,
another constant involved in asymptotic behavior, this time for the denominators of the convergents of continued fractions. (End update)
Question 2: Is there a closed-form expression for C(a,k) for other values {a,k} ?
-- End --
P.S. The next article will be on Ramanujan's beautiful continued fractions and its interesting connection to Platonic solids.
© Oct 2010, Tito Piezas III
You can email author at tpiezas@gmail.com