Chapter 7: Problem 8
Write the first five terms of the sequence. \(a_{n}=10+\frac{2}{n}+\frac{6}{n^{2}}\)
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Modeling Data The annual sales \(a_{n}\) (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) represents the year, with \(n=3\) corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report) (3,3844),(4,4267),(5,4492),(6,4814),(7,5079) (8,5213),(9,5289),(10,5682),(11,5958),(12,6171) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=3,4, \ldots, 12\) for the data. Graphically compare the points and the model. (b) Use the model to predict sales in the year 2008 .
Consider making monthly deposits of \(P\) dollars in a savings account at an annual interest rate \(r .\) Use the results of Exercise 106 to find the balance \(A\) after \(t\) years if the interest is compounded (a) monthly and (b) continuously. $$ P=\$ 75, \quad r=5 \%, \quad t=25 \text { years } $$
Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty\) and \(\sum b_{n}\) diverges, \(\sum a_{n}\) also diverges.
(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}}
Given two infinite series \(\sum a_{n}\) and \(\sum b_{n}\) such that \(\sum a_{n}\) converges and \(\sum b_{n}\) diverges, prove that \(\sum\left(a_{n}+b_{n}\right)\) diverges.
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