Question

How do you evaluate the continued fraction `f(e)=e-1+1/(e-1+1/(e-1+1/(e-1+1/(e-1+1/...` ?

Answer 1

`(e-1+sqrt(e^2-2e+5))/2=2.17752`, nearly.

Explanation:

Let `f(e)=x`, then `x=e-1+1/x`. Hence, `x^2=ex-x+1` or `x^2-x(e-1)-1=0` and using quadratic formula
`x=(e-1+-sqrt(e^2-2e+5))/2`
`x>e-1`

So, `x=(e-1+sqrt(e^2-2e+5))/2=2.17752,`nearly.