Explanation Given: The function is f ( x ) = x 2 e x . Result used: (1) Ratio test : Let ∑ k = 1 ∞ a k be an infinite series and let r = lim k → ∞ | a k + 1 a k | . If r < 1 , then the series ∑ k = 1 ∞ a k converges. Calculation: Obtain the power series for the function f ( x ) = x 2 e x . Multiply the series e x = ∑ k = 0 ∞ x k k ! by x 2 , x 2 e x = x 2 ∑ k = 0 ∞ x k k ! = ∑ k = 0 ∞ x k + 2 k ! Hence the power series representation of x 2 e x is ∑ k = 0 ∞ x k + 2 k ! . The interval of convergence of ∑ k = 0 ∞ ( x ) k + 2 k ! is computed as follows, Let a k = ( x ) k + 2 k ! . Then, a k + 1 = ( x ) k + 3 ( k + 1 ) !

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 11.2, Problem 74E

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To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

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Chapter 11.2, Problem 73E

Chapter 11.2, Problem 75E

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Chapter 11 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ < x < ∞ .

Solution Summary: The author calculates the power series for the given function and interval of convergence by using the exponential function centered at 0.

Videos

To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

Want to see the full answer?

Chapter 11 Solutions

The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ &lt; x &lt; ∞ .

Solution Summary: The author calculates the power series for the given function and interval of convergence by using the exponential function centered at 0.

Videos

To find: The power series for the given function and interval of convergence by using the exponential function centered at 0 is ex=∑k=0∞xkk!,for−∞<x<∞.

Want to see the full answer?

Chapter 11 Solutions

The power series for the given function and interval of convergence by using the exponential function centered at 0 is e x = ∑ k = 0 ∞ x k k ! , for − ∞ < x < ∞ .