Problem 84
A friend in your calculus class tells you that the following series converges because the terms are very small and approach 0 rapidly. Is your friend correct? Explain. \(\frac{1}{10,000}+\frac{1}{10,001}+\frac{1}{10,002}+\cdots\)
Problem 85
State the Direct Comparison Test and give an example of its use
Problem 85
Compound Interest Consider the sequence \(\left\\{A_{n}\right\\}\) whose \(n\) th term is given by \(A_{n}=P\left(1+\frac{r}{12}\right)^{n}\) where \(P\) is the principal, \(A_{n}\) is the account balance after \(n\) months, and \(r\) is the interest rate compounded annually. (a) Is \(\left\\{A_{n}\right\\}\) a convergent sequence? Explain. (b) Find the first 10 terms of the sequence if \(P=\$ 9000\) and \(r=0.055\)
Problem 85
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{n 7^{n}}{n !} $$
Problem 85
In Exercises 85 and \(86,\) (a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1+x+x^{2}+x^{3}+\cdots $$
Problem 86
(a) find the common ratio of the geometric series, \((b)\) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums \(S_{3}\) and \(S_{5} .\) What do you notice? $$ 1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\cdots $$
Problem 86
Inflation If the rate of inflation is \(4 \frac{1}{2} \%\) per year and the average price of a car is currently \(\$ 16,000,\) the average price after \(n\) years is \(P_{n}=\$ 16,000(1.045)^{n}\) Compute the average prices for the next 5 years.
Problem 86
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{\ln n}{n^{2}} $$
Problem 86
State the Limit Comparison Test and give an example of its use.
Problem 87
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. $$ \frac{\text { Function }}{f(x)=3\left[\frac{1-(0.5)^{x}}{1-0.5}\right]} \frac{\text { Series }}{\sum_{n=0}^{\infty} 3\left(\frac{1}{2}\right)^{n}} $$
