Limit of ratio of nested radicals

[garrus]

Homework Statement

I have a sequence a_n+1=sqrt(2+a_n) ,with a₀ = sqrt(2)
which leades to nested radicals.

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

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

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?

 

[uart]

Any hints?

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

 

[garrus]

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