Chapter 7: Problem 113
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(0.75=0.749999 \ldots \ldots\)
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A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).
Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty}(1.075)^{n} $$
Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).
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 .
Show that the series \(\sum_{n=1}^{\infty} a_{n}\) can be written in the telescoping form \(\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_{n}\right)\right]\) where \(S_{0}=0\) and \(S_{n}\) is the \(n\) th partial sum.
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