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Chapter 9: Problem 25

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. $$f(x)=\ln x, a=3$$

Short Answer

Expert verified
Answer: The first four non-zero terms of the Taylor series are: $$ f(x) \approx \ln(3) + \frac{1}{3}(x-3) - \frac{1}{18}(x-3)^2 + \frac{1}{27}(x-3)^3 \cdots $$ The power series expressed using summation notation: $$ f(x) \approx \ln(3) + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3^n n!}(x-3)^n $$

Step by step solution

01

Find the first four derivatives of the function

For this step, we'll find the first four derivatives of the given function \(f(x) = \ln x\) with respect to x: 1. First derivative: \(f'(x) = \frac{1}{x}\) 2. Second derivative: \(f''(x) = -\frac{1}{x^2}\) 3. Third derivative: \(f'''(x) = \frac{2}{x^3}\) 4. Fourth derivative: \(f^{(4)}(x) = -\frac{6}{x^4}\)
02

Evaluate the derivatives at the center point \(a=3\)

Now we'll evaluate each of the derivatives we found in step 1 at the center point \(a=3\): 1. \(f'(3) = \frac{1}{3}\) 2. \(f''(3) = -\frac{1}{3^2} = -\frac{1}{9}\) 3. \(f'''(3) = \frac{2}{3^3} = \frac{2}{27}\) 4. \(f^{(4)}(3) = -\frac{6}{3^4} = -\frac{6}{81}\)
03

Find the first four non-zero terms of the Taylor series

We'll now use the Taylor series formula to find the first four non-zero terms with the derivatives evaluated at the center point: $$ f(x) \approx f(3) + f'(3)(x-3) + \frac{f''(3)}{2!}(x-3)^2 + \frac{f'''(3)}{3!}(x-3)^3 + \cdots $$ Plug in the values we found in step 2: $$ f(x) \approx \ln(3) + \frac{1}{3}(x-3) - \frac{1}{9}\frac{(x-3)^2}{2} + \frac{2}{27}\frac{(x-3)^3}{3!} \cdots$$ Simplify the terms: $$ f(x) \approx \ln(3) + \frac{1}{3}(x-3) - \frac{1}{18}(x-3)^2 + \frac{1}{27}(x-3)^3 \cdots $$
04

Write the power series using summation notation

Now we'll write the power series as a summation, using the pattern of terms we found in step 3: $$ f(x) \approx \ln(3) + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3^n n!}(x-3)^n $$ Note that the first term \(\ln(3)\) is separate from the summation, since the series begins from the first degree term. Finally, here's our result: a. The first four non-zero terms of the Taylor series are: $$ f(x) \approx \ln(3) + \frac{1}{3}(x-3) - \frac{1}{18}(x-3)^2 + \frac{1}{27}(x-3)^3 \cdots $$ b. The power series expressed using summation notation: $$ f(x) \approx \ln(3) + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3^n n!}(x-3)^n $$

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Most popular questions from this chapter

An essential function in statistics and the study of the normal distribution is the error function $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ a. Compute the derivative of erf \((x)\). b. Expand \(e^{-t^{2}}\) in a Maclaurin series, then integrate to find the first four nonzero terms of the Maclaurin series for erf. c. Use the polynomial in part (b) to approximate erf \((0.15)\) and erf \((-0.09)\). d. Estimate the error in the approximations of part (c).

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