91Ó°ÊÓ

The sequence \((1 / \sqrt{n})\) has limit 0. For each of \(\epsilon=0.01,0.001,0.0001\), determine an integer \(N\) with the property that \(|(1 / \sqrt{n})-0|<\epsilon\) for all \(n>N\)

Show that the sequence \(\left(1 / n^{k}\right)_{n \in \mathrm{N}}\) is convergent if and only if \(k \geq 0\), and that the limit is 0 for all \(k>0\).

Determine the least value of \(N\) such that \(n /\left(n^{2}+1\right)<0.0001\) for all \(n \geq N\)

Determine the least value of \(N\) such that \(n^{2}+2 n \geq 9999\) for all \(n>N\)

Give a formal definition of the statement \(\left(a_{n}\right) \rightarrow-\infty\)

Let \(a_{1}=0, a_{2}=3\), and, for all \(n \geq 3\) let $$ a_{n}=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right) $$ (The sequence \(\left(a_{n}\right)\) is said to be defined recursively.) By induction on \(n\), show that, for all \(n \geq 2\), $$ a_{n}=2+4\left(-\frac{1}{2}\right)^{n} $$ and deduce that \(\left(a_{n}\right) \rightarrow 2\).

Let \(\left(b_{n}\right)\) be a sequence with limit \(\beta\). Show that if \(B\) is an upper bound for \(\left(b_{n}\right)\), then \(\beta \leq B\).

Show that a sequence \(\left(a_{n}\right)\) is bounded if and only if it is bounded above and below.

Let \(\left(a_{n}\right)\) be a sequence with limit \(\alpha\), and define \(b_{n}=a_{n+1}(n=\) \(1,2, \ldots) .\) Show that \(\left(b_{n}\right) \rightarrow \alpha\).

Show that, if \(a_{n} \geq 0\) for all \(n \geq 1\) and if \(\left(a_{n}\right) \rightarrow L\), then \(L \geq 0\).