91Ó°ÊÓ

Establish the convergence and find the limits of the following sequences: (a) \(\left(\left(1+1 / n^{2}\right)^{n^{2}}\right)\). (b) \(\left((1+1 / 2 n)^{n}\right)\), (c) \(\left(\left(1+1 / n^{2}\right)^{2 n^{2}}\right)\). (d) \(\left((1+2 / n)^{n}\right)\).

Let \(\left(x_{n}\right)\) be a Cauchy sequence such that \(x_{n}\) is an integer for every \(n \in \mathbb{N}\). Show that \(\left(x_{n}\right)\) is. ultimately constant.

Let \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) be sequences of positive numbers such that \(\lim \left(x_{n} / y_{n}\right)=0\). (a) Show that if \(\lim \left(x_{n}\right)=+\infty\), then \(\lim \left(y_{n}\right)=+\infty\). (b) Show that if \(\left(y_{n}\right)\) is bounded, then \(\lim \left(x_{n}\right)=0\).

Let \(x_{1}:=a>0\) and \(x_{n+1}:=x_{n}+1 / x_{n}\) for \(n \in \mathbb{N}\). Determine if \(\left(x_{n}\right)\) converges or diverges.

Show directly that a bounded, monotone increasing sequence is a Cauchy sequence.

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) \((\sin \sqrt{n})\).

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum a_{n}^{2}\) always convergent? Either prove it or give a counterexample.

Prove that \(\lim \left(x_{n}\right)=0\) if and only if \(\lim \left(\left|x_{n}\right|\right)=0\). Give an example to show that the convergence of \(\left(\left|x_{n}\right|\right)\) need not imply the convergence of \(\left(x_{n}\right)\).

Determine the limits of the following. (a) \(\left((3 n)^{1 / 2 n}\right)\), (b) \(\left((1+1 / 2 n)^{3 n}\right)\).

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n}}\) always convergent? Either prove it or give a counterexample.