A Functional Continued Fraction ( FCF ) is #exp_(cf)(a;a;a)=a^(a+a/a^(a+a/a^(a+...))), a 1#. Choosing #a=pi#, how do you prove that the 17-sd value of the FCF is 39.90130307286401?

Question

A Functional Continued Fraction ( FCF ) is #exp_(cf)(a;a;a)=a^(a+a/a^(a+a/a^(a+...))), a > 1#. Choosing #a=pi#, how do you prove that the 17-sd value of the FCF is 39.90130307286401?

Nathan Axel · Accepted Answer

See details in explanation for the derivation. Some Socratic graphs are now included for graphical verification.

Let #y = exp_(cf)(pi;pi;pi)=pi^(pi+pi/(pi^(pi+pi/(pi^(pi+...)))#. Then, The implicit form for this FCF value y is #y=pi^( pi + pi /y ) = pi^( pi( 1 + 1/y ) )#. A discrete anolog for approximating y is the nonlinear difference equation #y_n = pi^( pi ( 1 + 1/y_(n-1))# Adopting this for iteration, with starter value #y_0=pi^pi#, and making 15 iterations in long precision arithmetic,. #y=y_15=39.901130307206401#, nearly, with the forward difference #Deltay_14=y_15-y_14=0#, for this 17-sd precision. Here, 0 means smallness of order #10^(-18)#. Scaled local graphs, for cross check: Use #y = exp_(cf)(x;x;x)=x^(x(1+1/y))# for the graph x-range encloses #pi# for each of the two graphs. y-ranges are appropriate, for precision levels. The first is for higher precision. Read y against x = #pi# graph{y-x^(x(1+1/y))=0 [3.141592 3.141593 39.9011 39.90115]} graph{y-x^(x(1+1/y))=0 [1.6 4 0 60]}

Jace Anderson · Answer

undefined To prove that the 17-sd value of the Functional Continued Fraction (FCF) with $ a = \pi $ is $ 39.90130307286401 $, you can use numerical methods or software packages specifically designed for calculating continued fractions to iteratively compute the value of the expression. This approach will provide the desired result with high precision.

A Functional Continued Fraction ( FCF ) is #exp_(cf)(a;a;a)=a^(a+a/a^(a+a/a^(a+...))), a &gt; 1#. Choosing #a=pi#, how do you prove that the 17-sd value of the FCF is 39.90130307286401?

A Functional Continued Fraction ( FCF ) is #exp_(cf)(a;a;a)=a^(a+a/a^(a+a/a^(a+...))), a > 1#. Choosing #a=pi#, how do you prove that the 17-sd value of the FCF is 39.90130307286401?