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Chapter 3: Problem 12
If \(x_{1}>0\) and \(x_{n+1}:=\left(2+x_{n}\right)^{-1}\) for \(n \geq 1\), show that \(\left(x_{n}\right)\) is a contractive sequence. Find the limit.
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Investigate the convergence or the divergence of the following sequences: (a) \(\left(\sqrt{n^{2}+2}\right)\). (b) \(\left(\sqrt{n} /\left(n^{2}+1\right)\right)\). (c) \(\left(\sqrt{n^{2}+1} / \sqrt{n}\right)\), (d) \((\sin \sqrt{n})\).
Show directly from the definition that the following are Cauchy sequences. (a) \(\left(\frac{n+1}{n}\right)\). (b) \(\left(1+\frac{1}{2 !}+\cdots+\frac{1}{n !}\right)\).
Let \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) be sequences of positive numbers such that \(\lim \left(x_{n} / y_{n}\right)=+\infty\), (a) Show that if \(\lim \left(y_{n}\right)=+\infty\), then \(\lim \left(x_{n}\right)=+\infty\). (b) Show that if \(\left(x_{n}\right)\) is bounded, then \(\lim \left(y_{n}\right)=0\).
Show that if \(\left(x_{n}\right)\) is unbounded, then there exists a subsequence \(\left(x_{n_{k}}\right)\) such that \(\lim \left(1 / x_{n_{2}}\right)=0\).
Apply Theorem \(3.2: 11\) to the following sequences, where \(a, b\) satisfy
\(0
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