Find sqrt( 2 + sqrt( 2 + sqrt( 2 + )))

1. Oct 18, 2008

sara_87

Find the limit as n tends to infinity;

sqrt(2), sqrt(2+sqrt(2)), sqrt(2+sqrt(2+sqrt(2))), etc... n times

2. Oct 18, 2008

HallsofIvy

Re: limits

If an is the nth value, then $a_{n+1}= \sqrt{2+ a_n}$

IF the sequence {an} converges (you will need to prove that, perhaps by proving that it is an increasing sequence and has an upper bound), call the limit A.
Then we must have $\lim_{n\rightarrow \infty}a_{n+1}= \sqrt{2+ \lim_{n\rightarrow \infty}a_n}$ so $A= \sqrt{2+ A}$.

3. Oct 19, 2008

sara_87

Re: limits

Thank you very much.
All is clear.

4. Oct 20, 2008

sara_87

Re: limits

Sorry i have a small problem

How come A=sqrt(2+A) ???
A=limit of an not a(n+1)
so Limit a(n+1)=sqrt(2+A)
No?