Chapter 7: Problem 56
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+3)} $$
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Let \(\sum a_{n}\) be a convergent series, and let \(R_{N}=a_{N+1}+a_{N+2}+\cdots\) be the remainder of the series after the first \(N\) terms. Prove that \(\lim _{N \rightarrow \infty} R_{N}=0\).
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 \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).
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. $$ P=\$ 75, \quad r=5 \%, \quad t=25 \text { years } $$
Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 6\left(\frac{4}{5}\right)^{n} $$
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. $$ \frac{\text { Function }}{f(x)=2\left[\frac{1-(0.8)^{x}}{1-0.8}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}} $$
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