The Golden Ratio φ

φ calculated with Fibonacci

The Golden Ratio φ "phi" can be calculated with a recursive definition. It is the limit of the quotient of the fibonacci sequence (fn)n∈N defined by
f0 = f1 = 0, fn = fn-1 + fn-2 for n≥2
φ = limn→∞ (fn/fn-1)
After more than 60000 steps the Golden Ratio appears to be:
φ =

Which has at least 9000 digits of exact precision (in fact it's full precision). The sequence calculating it is converging to

(√5+1)/ 2 = (Sqrt(5)+1) / 2.
Which can be proven analytically to be the exact value of φ. By using generating functions, it is possible to prove a closed form for the fibonacci sequence (by using of the golden ratio)
fn = (φn - ψn) / √5, with
φ = (1+√5) / 2
ψ = (1-√5) / 2