The Fibonacci sequence is divergent and it's terms tend to infinity. So, every term in the Fibonacci sequence (for n>2 ) is greater then it's predecessor.
Does Fibonacci series converge?
A sequence x = (xk) is said to be Fibonacci statistically convergent (or F ̂ -statistically convergent) if there is a number L such that, for every ϵ > 0, the set K ϵ ( F ˆ ) : = { k ≤ n : | F ˆ x k − L | ≥ ϵ } has natural density zero, i.e., d ( K ϵ ( F ˆ ) ) = 0 .
Does the golden ratio converge?
and if you calculate a few more terms of this sequence you will find that it rapidly converges to \phi giving the value to six significant figures, 1.61803, in just thirteen steps and giving more accuracy with more steps.
