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Chapter 9: Problem 6
Write the first five terms of the sequence. $$ a_{n}=\frac{2 n}{n+3} $$
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Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Every decimal with a repeating pattern of digits is a rational number.
Arithmetic-Geometric Mean Iet \(a_{0}>b_{0}>0\). I.et \(a_{1}\) be the arithmetic mean of \(a_{0}\) and \(b_{0}\) and let \(b_{1}\) be the geometric mean of \(a_{0}\) and \(b_{0}\). \(a_{1}=\frac{a_{0}+b_{0}}{2} \quad\) Arithmetic mean \(b_{1}=\sqrt{a_{0} b_{0}} \quad\) Geometric mean Now define the sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) as follows. \(a_{n}=\frac{a_{n-1}+b_{n-1}}{2} \quad b_{n}=\sqrt{a_{n-1} b_{n-1}}\) (a) Let \(a_{0}=10\) and \(b_{0}=3 .\) Write out the first five terms of \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\} .\) Compare the terms of \(\left\\{b_{n}\right\\} .\) Compare \(a_{n}\) and \(b_{n}\). What do you notice? (b) Use induction to show that \(a_{n}>a_{n+1}>b_{n+1}>b_{n}\), for \(a_{0}>b_{0}>0\) (c) Explain why \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are both convergent. (d) Show that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}\).
The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}\), where \(a_{1}=1\) and \(a_{2}=1\). (a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\). (b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).
Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty}(-1)^{n} x^{2 n} $$
(a) Show that \(\int_{1}^{n} \ln x d x<\ln (n !)\) for \(n \geq 2\).
(b) Draw a graph similar to the one above that shows
\(\ln (n !)<\int_{1}^{n+1} \ln x d x\)
(c) Use the results of parts (a) and (b) to show that
\(\frac{n^{n}}{e^{n-1}}
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