91Ó°ÊÓ

Use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. $$ \arctan (0.4) \approx 0.4-\frac{(0.4)^{3}}{3} $$

State the Limit Comparison Test and give an example of its use.

(a) Write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. $$ 0.0 \overline{75} $$

Use power series to approximate the value of the integral with an error of less than 0.0001. (In Exercises 55 and 56 , assume that the integrand is defined as 1 when \(x=0\).) \(\int_{0}^{1 / 2} \frac{\arctan x}{x} d x\)

Write a power series that has the indicated interval of convergence. Explain your reasoning. (a) \((-2,2)\) (b) \((-1,1]\) (c) \((-1,0)\) (d) \([-2,6)\)

Define the binomial series. What is its radius of convergence?

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} x^{n}, \quad-1

Show that the Maclaurin series of the function \(g(x)=\frac{x}{1-x-x^{2}}\) is \(\sum_{n=1}^{\infty} F_{n} x^{n}\) where \(F_{n}\) is the \(n\) th Fibonacci number with \(F_{1}=F_{2}=1\) and \(F_{n}=F_{n-2}+F_{n-1}\), for \(n \geq 3 .\) (Hint: Write \(\frac{x}{1-x-x^{2}}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots\) and multiply each side of this equation by \(1-x-x^{2}\).)

Probability 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).

Salary You accept a job that pays a salary of \(\$ 40,000\) for the first year. During the next 39 years you receive a \(4 \%\) raise each year. What would be your total compensation over the 40-year period?