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Chapter 10: Problem 16

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\sin 3 x$$

Short Answer

Expert verified
Based on the given function \(f(x) = \sin 3x\), we found the first four nonzero terms of its Maclaurin series as: $$3x - \frac{27}{3!}x^3$$ The power series in summation notation is: $$\sin 3x = \sum_{n=0}^{\infty} (-1)^n \frac{(3x)^{2n+1}}{(2n+1)!}$$ The interval of convergence for this series is: $$-\infty < x < \infty$$

Step by step solution

01

Find the first four nonzero terms

To find the first four nonzero terms, we need to compute the derivatives of \(f(x) = \sin 3x\) until the fourth derivative. Then, we will evaluate each derivative at x = 0: 1. \(f(x) = \sin 3x\) 2. \(f'(x) = 3 \cos 3x\) 3. \(f''(x) = -9 \sin 3x\) 4. \(f'''(x) = -27 \cos 3x\) Now, evaluate these at x = 0: 1. \(f(0) = \sin(0) = 0\) 2. \(f'(0) = 3 \cos(0) = 3\) 3. \(f''(0) = -9 \sin(0) = 0\) 4. \(f'''(0) = -27 \cos(0) = -27\) The first four nonzero terms of the Maclaurin series for \(f(x) = \sin 3x\) are: $$3x - \frac{27}{3!}x^3$$
02

Write the power series using summation notation

To write the power series using summation notation, recall that the sine function has a Maclaurin series that converges for all x: $$\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$$ We can use this general formula to express \(f(x)=\sin 3x\) in terms of summation notation: $$\sin 3x = \sum_{n=0}^{\infty} (-1)^n \frac{(3x)^{2n+1}}{(2n+1)!}$$
03

Determine the interval of convergence

Since the Maclaurin series for the sine function converges for all x, the interval of convergence of the series for \(f(x) = \sin 3x\) is the same. Thus, the interval of convergence is: $$-\infty < x < \infty$$

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Most popular questions from this chapter

Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is $$J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k}.$$ a. Write out the first four terms of \(J_{0}\) b. Find the radius and interval of convergence of the power series for \(J_{0}\) c. Differentiate \(J_{0}\) twice and show (by keeping terms through \(x^{6}\) ) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\)

Best expansion point Suppose you wish to approximate \(e^{0.35}\) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or \(\ln 2 ?\) Use a calculator for numerical experiments and check for consistency with Theorem 2. Does the answer depend on the order of the polynomial?

If the power series \(f(x)=\sum c_{k} x^{k}\) has an interval of convergence of \(|x|

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\left(1+x^{2}\right)^{-2 / 3}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{1}{x^{4}+2 x^{2}+1}$$
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