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Chapter 10: Problem 24

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. $$f(x)=1 / x, a=2$$

Short Answer

Expert verified
Answer: The first four nonzero terms of the Taylor series are \(\frac{1}{2} - \frac{1}{4}(x-2) + \frac{1}{4}(x-2)^2 - \frac{3}{8}(x-2)^3\), and the power series using summation notation is: $$f(x) \approx \sum_{n=1}^\infty \frac{(-1)^{n+1}(x-2)^{n-1}}{2^{n+1}n}$$

Step by step solution

01

Determine the First Four Derivatives

Starting with the original function \(f(x) =\frac{1}{x}\), we will find the first through the fourth derivatives of the function. First derivative: $$f'(x) = -\frac{1}{x^2}$$ Second derivative: $$f''(x) = \frac{2}{x^3}$$ Third derivative: $$f^{(3)}(x) = -\frac{6}{x^4}$$ Fourth derivative: $$f^{(4)}(x) = \frac{24}{x^5}$$
02

Evaluate Derivatives at a = 2

Next, we need to substitute \(a=2\) into each of the derivatives. $$f(2)=\frac{1}{2}$$ $$f'(2) = -\frac{1}{4}$$ $$f''(2) = \frac{2}{8}$$ $$f^{(3)}(2) = -\frac{6}{16}$$ $$f^{(4)}(2) = \frac{24}{32}$$
03

Calculate the Taylor Series Terms

Now, we will use the Taylor series formula with the derivatives of f evaluated at \(a=2\) to find the first four nonzero terms of the series: $$f(x) \approx f(2) + f'(2)(x-2) + \frac{f''(2)(x-2)^2}{2!} + \frac{f^{(3)}(2)(x-2)^3}{3!} + \frac{f^{(4)}(2)(x-2)^4}{4!}$$ $$f(x) \approx \frac{1}{2} - \frac{1}{4}(x-2) + \frac{1}{4}(x-2)^2 - \frac{3}{8}(x-2)^3 + \frac{3}{4}(x-2)^4$$
04

Write the Power Series using Summation Notation

The power series can be represented using summation notation as follows: $$f(x) \approx \sum_{n=1}^\infty \frac{(-1)^{n+1}(x-2)^{n-1}}{2^{n+1}n}$$ So, the first four nonzero terms of the Taylor series for this function centered at \(a=2\) are \(\frac{1}{2} - \frac{1}{4}(x-2) + \frac{1}{4}(x-2)^2 - \frac{3}{8}(x-2)^3\), and the power series using summation notation is: $$f(x) \approx \sum_{n=1}^\infty \frac{(-1)^{n+1}(x-2)^{n-1}}{2^{n+1}n}$$

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Most popular questions from this chapter

Recall that the Taylor series for \(f(x)=1 /(1-x)\) about 0 is the geometric series \(\sum_{k=0}^{\infty} x^{k} .\) Show that this series can also be found as a case of the binomial series.

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

Find the next two terms of the following Taylor series. $$\sqrt{1+x}: 1+\frac{1}{2} x-\frac{1}{2 \cdot 4} x^{2}+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6} x^{3}-\cdots$$

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

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. $$f(x)=\sqrt{1-x^{2}}$$
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