91Ó°ÊÓ

Differential Equation In Exercises \(59-64,\) show that the function represented by the power series is a solution of the differential equation. $$y=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !}, \quad y^{\prime \prime}+y=0$$

Approximating an Integral In Exercises \(63-70\) , use a power series to approximate the value of the definite integral with an error of less than \(0.0001 .\) (In Exercises 65 and \(67,\) assume that the integrand is defined as 1 when \(x=0 . )\) $$\int_{0}^{1} \cos x^{2} d x$$

Approximating an Integral In Exercises \(63-70\) , use a power series to approximate the value of the definite integral with an error of less than \(0.0001 .\) (In Exercises 65 and \(67,\) assume that the integrand is defined as 1 when \(x=0 . )\) $$\int_{0}^{1 / 2} \frac{\arctan x}{x} d x$$

$$\begin{array}{l}{\text { Using a Power Series } \text { Let }} \\ {g(x)=1+2 x+x^{2}+2 x^{3}+x^{4}+\cdots} \\ {\text { where the coefficients are } c_{2 n}=1 \text { and } c_{2 n+1}=2 \text { for } n \geq 0 \text { . }} \\ {\text { (a) Find the interval of convergence of the series. }} \\ {\text { (b) Find an explicit formula for } g(x) \text { . }}\end{array}$$

Probability In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .

Finding a Maclaurin Series Find the Maclaurin series for \(f(x)=\ln \frac{1+x}{1-x}\) and determine its radius of convergence. Use the first four terms of the series to approximate ln 3 .

The sphereflake shown below is a computer-generated fractal that was created by Eric Haines. The radius of the large sphere is 1. To the large sphere, nine spheres of radius \(\frac{1}{3}\) are attached. To each of these, nine spheres of radius \(\frac{1}{9}\) are attached. This process is continued infinitely. Prove that the sphereflake has an infinite surface area.

Evaluating a Binomial Coefficient In Exercises \(85-88\) evaluate the binomial coefficient using the formula $$\left( \begin{array}{l}{k} \\ {n}\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}$$ where \(k\) is a real number, \(n\) is a positive integer, and \(\left( \begin{array}{l}{k} \\\\{0}\end{array}\right)=1\). $$\left( \begin{array}{c}{-5} \\ {6}\end{array}\right)$$

Think About It What can you conclude about the convergence or divergence of \(\Sigma a_{n}\) using the Ratio Test when \(a_{n}\) is a rational function of \(n ?\) Explain.