# Prove this sequence converges

• Feb 6th 2010, 11:16 AM
Em Yeu Anh
Prove this sequence converges
Suppose that $\displaystyle a_1 = 1$ and $\displaystyle a_n = \sqrt{2+a_{n-1}}$ for all $\displaystyle n \geq 2.$ Prove that this sequence converges and find its limit.

I was hoping that I could avoid induction on this one but I don't know any other method to prove this.
• Feb 6th 2010, 12:12 PM
tonio
Quote:

Originally Posted by Em Yeu Anh
Suppose that $\displaystyle a_1 = 1$ and $\displaystyle a_n = \sqrt{2+a_{n-1}}$ for all $\displaystyle n \geq 2.$ Prove that this sequence converges and find its limit.

I was hoping that I could avoid induction on this one but I don't know any other method to prove this.

Yes , induction is the key here. Prove that this is monotone increasing sequence and that $\displaystyle a_n\leq 2\,\,\,\forall n\in\mathbb{N}$

Tonio
• Feb 6th 2010, 12:18 PM
felper
You can prove that the sequence is crecent and is bounded. To calculate the limit, you can use that the subsequences of a sequences converges to the same limit. Then, $\displaystyle L=\sqrt{2+L}$