Chapter 7: Problem 48
(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{9} $$
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In Exercises 87 and 88 , use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series. $$ \frac{\text { Function }}{f(x)=3\left[\frac{1-(0.5)^{x}}{1-0.5}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 3\left(\frac{1}{2}\right)^{n}} $$
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.
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.
The random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{3}\left(\frac{2}{3}\right)^{n} $$
Determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{n}{\ln n} $$
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