# Homework Help: Limit of ratio of nested radicals

1. Aug 7, 2012

### garrus

1. The problem statement, all variables and given/known data
I have a sequence an+1=sqrt(2+an) ,with a0 = sqrt(2)

I am asked to show that for n approaching infinity:
1) the sequence converges to 2, and that

2) lim {(an+1 -2) / (an -2)} = 1/4

3. The attempt at a solution
1)I have proven the convergence to 2.
2) for the 2nd limit, i tried de L'Hospital's rule, since we have 0 / 0 , but can't figure out exactly how to find the derivative of an (infinite) nested roots.
- i tried also square differences , but still get to 0/0.

Any hints?

2. Aug 7, 2012

### uart

Yep. Factorize $(a_{n+1}^2 - 4)$ and compare with the same expression derived from your recurrence relation.

3. Aug 7, 2012

### garrus

Got it.
Thanks for your help, should've noticed that =[