Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=2(9-x)^{\frac{1}{2}}, \quad n=3 $$

Step-by-Step Solution

Answer

The third degree Taylor polynomial for the function \(f(x)\) is \(P_3(x) = 6 - x + \frac{1}{8} x^2 - \frac{1}{48} x^3\), and the radius of convergence for the corresponding Maclaurin series is \(R=1\)

Step 1: Find the Taylor Polynomial

In order to find the Taylor polynomial of degree 3 for the function \(f(x)=2(9-x)^{\frac{1}{2}}\), the first three derivatives of the function need to be computed at \(x=0\). The general formula for the nth derivative at \(x=0\) is \[f^{(n)}(0) = \frac{n!}{2} \cdot (-1)^n \cdot (9-x)^{\frac{1}{2}-n}\]Substituting \(n=0,1,2,3\) yields \(f(0)=6\), \(f'(0)=-1\), \(f''(0)=\frac{1}{4}\) and \(f'''(0)=-\frac{1}{8}\). Therefore, the third degree Taylor polynomial obtained from these results is\[P_3(x) = 6 - x + \frac{1}{4} \frac{x^2}{2!} - \frac{1}{8} \frac{x^3}{3!}\]

Step 2: Find the Radius of Convergence

The ratio test can be used to determine the radius of convergence (\(R\)) for the Maclaurin series. In the ratio test, if the absolute value of the ratio of the \(n+1\)th term to the nth term is less than 1, then the series converges. Simplifying this inequality provides the radius of convergence. For the given function, the Maclaurin series is a binomial series that converges for \(-1

Step 3: Conclusion

The third degree Taylor polynomial for the function \(f(x)\) is \(P_3(x) = 6 - x + \frac{1}{8} x^2 - \frac{1}{48} x^3\). The radius of convergence for the corresponding Maclaurin series is \(R=1\)

Key Concepts

Binomial Series

The binomial series provides a way to expand expressions of the form \( (1+x)^n \), where \( n \) is any real number. While in school, you may have learned about the binomial theorem, which is a special case applicable when \( n \) is a non-negative integer. The binomial series, however, is more general and even applies when \( n \) is not an integer.

Radius of Convergence

In the world of infinite series, the radius of convergence is a measure that tells us where a power series like the Maclaurin or Taylor series converges. To visualize it, imagine drawing a circle on the number line with the center at the expansion point of the series.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at \( x = 0 \). It's a powerful tool because it allows us to represent many complex functions as infinite polynomials, using derivatives to find the coefficients.

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Other exercises in this chapter

Problem 20

Suppose that \(a_{k}=f(k)\) for \(k=1,2,3, \ldots\), where \(f(x)\) is positive, decreasing, and continuous on \([1, \infty)\). Put the following expressions in

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Problem 20

Find the Maclaurin series for \(\ln (2+x)\) along with its radius of convergence.

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Problem 20

Find the fth degree Taylor polynomial for \(\sqrt{x}\) centered at \(x=9\).

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Problem 21

Explain why the hypothesis that \(f(x)\) is decreasing is important in the Integral Test.

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Key Concepts

Binomial Series

The binomial series provides a way to expand expressions of the form \( (1+x)^n \), where \( n \) is any real number. While in school, you may have learned about the binomial theorem, which is a special case applicable when \( n \) is a non-negative integer. The binomial series, however, is more general and even applies when \( n \) is not an integer.

Radius of Convergence

In the world of infinite series, the radius of convergence is a measure that tells us where a power series like the Maclaurin or Taylor series converges. To visualize it, imagine drawing a circle on the number line with the center at the expansion point of the series.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series centered at \( x = 0 \). It's a powerful tool because it allows us to represent many complex functions as infinite polynomials, using derivatives to find the coefficients.