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10th Edition

Ron Larson

Publisher: Cengage Learning

ISBN: 9781305860919

Chapter 10.4, Problem 32E

Textbook Problem

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**Finding Taylor and Maclaurin Series** In Exercises 25–32, apply Taylor’s Theorem to find the power series for the function centered at c. Then find the radius of convergence. *See Examples 4 and 5*.

To determine

**To calculate:** The power series for the function

**Given Information:**

The provided function

**Formula used:**

The simple power rule of derivative

The power series of a function *f* centered at *c* from Taylor’s theorem is

Ratio test:

For an infinite series

a) If

b) If

c) If

Convergence of a Power Series:

If a power series is centered at a constant *c*, then either of the following condition is true.

a) The radius of convergence is *c*.

b) The radius of convergence

c) The radius of convergence is

The chain rule for derivative of composite function is,

**Calculation:**

Consider the function

Substitute 4 for *x* in the function

Therefore, the value of

Use the simple power rule to differentiate the provided function.

Substitute 4 for *x* in the above first derivative function.

Therefore, the value of

Use the simple power rule to differentiate the first derivative

Substitute 4 for *x* in the above second derivative function.

Use the simple power rule to differentiate the second derivative

Substitute 4 for *x* in the above third derivative function.

Use the simple power rule to differentiate the third derivative

Substitute 4 for *x* in the above fourth derivative function.

From the above derivative pattern, the *n*th derivative of the provided function in general for

Substitute 4 for *x* in the above nth derivative function

Calculus: An Applied Approach (MindTap Course List)

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Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding Taylor and Maclaurin Series In Exercises...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Finding a Maclaurin Series In Exercises 33-36, use...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Using the Basic list of Power Series In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding the Radius of Convergence In Exercises...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Finding Power Series In Exercises 5356, find the...Ch. 10.4 - Think About It Consider the geometric series...Ch. 10.4 - HOW DO YOU SEE IT? Match the graph of the first 10...Ch. 10.5 - Find the 12th-degree Taylor polynomial for...Ch. 10.5 - Use the fourth-degree Taylor polynomial from...Ch. 10.5 - Checkpoint 3 Worked-out solution available at...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 16, find the power series for the...Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - In Exercises 710, evaluate the definite integral....Ch. 10.5 - Using Taylor Polynomials In Exercises 1 and 2, use...Ch. 10.5 - Using Taylor Polynomials In Exercises 1 and 2, use...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 314, find...Ch. 10.5 - Finding Taylor Polynomials In Exercises 15 and 16,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 15 and 16,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 17 and 18,...Ch. 10.5 - Finding Taylor Polynomials In Exercises 17 and 18,...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Matching In Exercises 19-22, match the Taylor...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Using a Taylor Polynomial Approximation In...Ch. 10.5 - Finding the Maximum Error In Exercises 27 and 28,...Ch. 10.5 - Finding the Maximum Error In Exercises 27 and 28,...Ch. 10.5 - Using the Maximum Error In Exercises 29 and 30,...Ch. 10.5 - Using the Maximum Error In Exercises 29 and 30,...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Approximating a Definite Integral In Exercises...Ch. 10.5 - Comparing Taylor Polynomials Consider the...Ch. 10.5 - HOW DO YOU SEE IT? 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