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 φ_n 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 = x_n 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-x_n)(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

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