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Chapter 10: Problem 23

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=1 / x, a=1$$

Short Answer

Expert verified
The first four nonzero terms of the Taylor series are: $$f(x) \approx 1 - (x-1) + 2(x-1)^2 - 6(x-1)^3$$ The power series in summation notation is: $$f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n (n!)}{x^{n+1}}$$

Step by step solution

01

Find the derivatives of the function

Find the first few derivatives of \(f(x)=\frac{1}{x}\) at \(x=1\) to get the Taylor series coefficients. $$f(x)=\frac{1}{x} \Rightarrow f'(x)=-\frac{1}{x^2} \Rightarrow f''(x)=\frac{2}{x^3} \Rightarrow f^{(3)}(x)=-\frac{6}{x^4} \Rightarrow f^{(4)}(x)=\frac{24}{x^5}$$
02

Evaluate derivatives at \(a=1\)

Compute the first, second, third, and fourth derivatives of the function at the point \(a=1\) to get the coefficients of the Taylor series. $$f(1)=1, \quad f'(1)=-1, \quad f''(1)=2, \quad f^{(3)}(1)=-6, \quad f^{(4)}(1)=24$$
03

Write the first four nonzero terms of the Taylor series

Write down the first four Taylor series terms using the coefficients found in the previous steps. $$f(x) \approx 1 - (x-1) + 2(x-1)^2 - 6(x-1)^3$$
04

Write the power series using summation notation

Represent the series obtained in step 3 using summation notation. $$\sum_{n=0}^{\infty} \frac{(-1)^n (n!)}{x^{n+1}} = 1 - (x-1) + 2(x-1)^2 - 6(x-1)^3 + \cdots$$ Therefore, the first four nonzero terms of the Taylor series are: $$f(x) \approx 1 - (x-1) + 2(x-1)^2 - 6(x-1)^3$$ and the power series in summation notation is: $$f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n (n!)}{x^{n+1}}$$

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Most popular questions from this chapter

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=e^{-x}, a=0$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The Taylor polynomials for \(f(x)=e^{-2 x}\) centered at 0 consist of even powers only. b. For \(f(x)=x^{5}-1,\) the Taylor polynomial of order 10 centered at \(x=0\) is \(f\) itself. c. The \(n\) th-order Taylor polynomial for \(f(x)=\sqrt{1+x^{2}}\) centered at 0 consists of even powers of \(x\) only.

By comparing the first four terms, show that the Maclaurin series for \(\sin ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\sin x,\) (b) by using the identity \(\sin ^{2} x=(1-\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.

Identify the functions represented by the following power series. $$\sum_{k=0}^{\infty} 2^{k} x^{2 k+1}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}\left(\frac{x^{2}-1}{3}\right)^{k}$$
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