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Chapter 3: Problem 1
Let \(x_{1}:=8\) and \(x_{n+1}:=\frac{1}{2} x_{n}+2\) for \(n \in \mathbb{N}\). Show that \(\left(x_{n}\right)\) is bounded and monotone. Find the limit.
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Let \(x_{n}:=1 / \ln (n+1)\) for \(n \in \mathbb{N}\) (a) Use the definition of limit to show that \(\lim \left(x_{n}\right)=0\). (b) Find a specific value of \(K(\varepsilon)\) as required in the definition of limit for each of (i) \(\varepsilon=1 / 2\), and (ii) \(\varepsilon=1 / 10\).
If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n} a_{n+1}}\) always convergent? Either prove it or give a counterexample.
Give an example of two divergent sequences \(X\) and \(Y\) such that: (a) their sum \(X+Y\) converges, (b) their product \(X Y\) converges.
The polynomial equation \(x^{3}-5 x+1=0\) has a root \(r\) with \(0
List the first five terms of the following inductively defined sequences. (a) \(x_{1}:=1, \quad x_{n+1}=3 x_{n}+1\), (b) \(y_{1}:=2, \quad y_{n+1}=\frac{1}{2}\left(y_{n}+2 / y_{n}\right)\) (c) \(z_{1}:=1, \quad z_{2}:=2, \quad z_{n+2}:=\left(z_{n+1}+z_{n}\right) /\left(z_{n+1}-z_{n}\right)\) (d) \(s_{1}=3, \quad s_{2}:=5, \quad s_{n+2}:=s_{n}+s_{n+1}\).
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