Chapter 9.2, Problem 27ES

Let $\left\{a_n\right\}$ be the sequence defined recursively by $a_1=\sqrt{2}\text{and}{a}_{n+1}=\sqrt{2+a_n}\text{for}n\ge 1.$

(a) List the first three terms of the sequence.

(b) Show that $a_n<2\text{for}n\ge 1.$

(c) Show that $a_{n+1}^2-a_n^2=\left(2-a_n\right)\left(1+a_n\right)\text{for}n\ge 1.$

(d) Use the result in parts (b) and (c) to show that $\left\{a_n\right\}$ is a strictly increasing sequence.

(e) Show that $\left\{a_n\right\}$ converges and find its limit L.