Chapter 7: Problem 1
In Exercises \(1-6,\) find the first five terms of the sequence of partial sums. $$ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdot \cdot $$
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Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\Sigma a_{n}\) also converges.
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 \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$
(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}}
Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A sequence that converges to \(\frac{3}{4}\)
Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\) (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by \(b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1\) (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho .\) Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).
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