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

• Nov 18th 2012, 10:57 AM
lm1988
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?
• Nov 18th 2012, 04:27 PM
chiro
Re: Prove that the relation x_n = 2 + sqrt(x_(n-1) + 1) converges given that x_0>=2.
Hey lm1988.

Can you get an expansion for this series in terms of n's? (like a power series type expansion)?
• Nov 18th 2012, 04:44 PM
lm1988
Re: Prove that the relation x_n = 2 + sqrt(x_(n-1) + 1) converges given that x_0>=2.
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.
• Nov 18th 2012, 05:15 PM
chiro
Re: Prove that the relation x_n = 2 + sqrt(x_(n-1) + 1) converges given that x_0>=2.
Well what identities were you given?