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

[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

 

[HallsofIvy]

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

IF the sequence {a_n} 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 $A= \sqrt{2+ A}$.

 

[sara_87]

Thank you very much.
All is clear.

 

[sara_87]

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?