Homework 7: Monotone Sequences

Instructions. Assignments should be stapled and written clearly and legibly.

1. Prove that the sequence \((a_n)\) converges, where \(a_n = \dfrac{1\cdot 3\cdot 5\cdots (2n - 1) }{2\cdot 4 \cdot 6\cdots (2n)}\).
2. In this problem, we give an algorithm for computing \(\sqrt{2}\). Let \(a_1=2\), and define \begin{equation*} a_{n+1}=\frac{1}{2}\left(a_n+\frac{2}{a_n}\right), \text{ for }n\geq 1. \end{equation*}
   1. Prove that \(a_n^2\geq 2\) for all \(n\).
   2. Use part (a) and the recursive equation above to prove that \(a_n-a_{n+1}\geq 0\) for all \(n\).
   3. Conclude that the sequence \((a_n)\) converges.
   4. Prove that \(\lim\limits_{n\to \infty}a_n=\sqrt{2}\).
   5. Modify the sequence \((a_n)\) so that it converges to \(\sqrt{c}\). No formal proof is required for this part, but you should give a brief justification.
3. 1. Prove that if \(0\lt a\lt 2\), then \(a\lt \sqrt{2a} \lt 2\).
   2. Use part (a) to prove that the sequence \begin{equation*} \left(\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}},\ldots\right) \end{equation*} converges.
   3. Find the limit.