Converging Sequence: Find the Limit of t_n

[akoska]

Homework Statement

Let t1=sqrt(2)

Let t_(n+1)=sqrt(2+sqrt(t_n)) (it's a recursively defined series)

What does it converge to?

Homework Equations

The Attempt at a Solution

I calculated it out for some values andI get 1.8312 (approx), but I don't want to express it in decimals, and I want to know if there's a good way to do this.

 

[IMDerek]

Let's suppose it converges, then t_(n+1) and t_n will have a infinitesimally small difference as n approaches infinity, and you can consider them the same number. (aka the limit of this sequence)

 

[HallsofIvy]

I would have phrased it differently, but basically said the same thing. Since {t_{n+1[/b]} is exactly the same sequence as {tn}, just indexed differently, taking the limit as n goes to infinity gives the same value, say T, on both sides. Solve that equation for T. (That's not a trivial equation but there is one obvious root!)}