0014: Article 4 (Golden Ratio and Nested Radicals)

 
 
 

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 

 

 
 
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