Question

Sequence and Series

\text { Define the sequence }\left(a_{n}\right)_{n \geq 1} \text { recursively by } a_{1}=1 \text { and } a_{n+1}=\sqrt{2+a_{n}} \text { for all } n \geq 1

\text { i. Prove that for all } n \geq 1 \text { we have } 1 \leq a_{n} \leq 2

\text { ii. Prove that }\left(a_{n}\right)_{n \geq 1} \text { is a monotonic sequence. }

iii. Deduce that the sequence converges with limit 2.

iv. Show that for all n > 1 we have

2-a_{n} \leq \frac{1}{(2+\sqrt{3})^{n-1}}


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