Prove that the relation x_n = 2 + sqrt(x_(n-1) + 1) converges given that x_0>=2.

I know we can probably make a conjecture, that that if the number is low enough it is a strictly increasing function but converges, and if the number is high enough, it is a strictly decreasing function but converges. How can you prove it without numbers?

Hey lm1988.

Can you get an expansion for this series in terms of n's? (like a power series type expansion)?

This problem should not require a power series expansion as it was on a recent test I had to take wherein PS expansion was not among the required material for preparation. We had been taught about convergence and sequences.

Well what identities were you given?