Explanation Given: The function f ( x ) = e 2 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 is converges. Calculation: Obtain the power series for the function f ( x ) = e − x . Substitute 2 x for x in e x = ∑ k = 0 ∞ x k k ! , e 2 x = ∑ k = 0 ∞ ( 2 x ) k k ! Hence, the power series representation of e 2 x is ∑ k = 0 ∞ ( 2 x ) k k ! . The interval of convergence of ∑ k = 0 ∞ ( 2 x ) k k ! is computed as follows, Let a k = ( 2 x ) k k ! . Then, a k + 1 = ( 2 x ) k + 1 ( k + 1 ) ! . Use the Ratio test to compute the value | a k + 1 a k |

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

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Chapter 9.2, Problem 70E

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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 9.2, Problem 69E

Chapter 9.2, Problem 71E

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Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

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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 explains how to find the power series for the function f(x)=e2x

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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<∞.

Want to see the full answer?

Chapter 9 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 explains how to find the power series for the function f(x)=e2x

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 9 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 < ∞ .