# 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?

##### 1 Answer

#### Answer:

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

#### Explanation:

Let

implicit form for this FCF value y is

A discrete anolog for approximating y is the nonlinear difference

equation

Adopting this for iteration, with starter value

making 15 iterations in long precision arithmetic,.

with the forward difference

Here, 0 means smallness of order

Scaled local graphs, for cross check:

Use

x-range encloses

y-ranges are appropriate, for precision levels.

The first is for higher precision.

Read y against x =

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]}