(a) Explanation Given: The provided series is ∑ k = 0 ∞ ( − 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) and the error in the approximation of the sum of the series should be less than 0.0001 . Formula used: For a convergent alternating series the error in approximating the sum S by the partial sum S N is, | S − S N | ≤ a N + 1 , where a N + 1 is the first neglected term, provided the series satisfies the condition a n + 1 ≤ a n . Calculate: Consider the provided series, ∑ k = 0 ∞ ( − 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) First check whether the above series satisfies the condition a k + 1 ≤ a k or not. The k th term of the provided series is, a k = ( − 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) …… (1) Similarly, the ( k + 1 ) th term is, a k + 1 = ( 3 ) k + 1 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 3 ) On simplification, a k + 1 = ( 3 2 k + 3 ) ⋅ ( 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) …… (2) Substitute value from equation (1) to get, a k + 1 = ( 3 2 k + 3 ) ⋅ a k Now, for k ≥ 0 the term ( 3 2 ( k + 1 ) ) is less than or equal to one that is, ( 3 2 k + 3 ) ≤ 1 Therefore, a k + 1 ≤ a k Now, let the number of term required to approximate the sum of the provided series with an error less than 0 (b) To determine To calculate: The approximatesum of the series ∑ k = 0 ∞ ( − 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) with an error less than 0.0001 using graphing utility.

Bundle: Calculus: Early Transcendental Functions, Loose-leaf Version, 6th + WebAssign Printed Access Card for Larson/Edwards' Calculus: Early Transcendental Functions, 6th Edition, Multi-Term

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ISBN: 9781305714045

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 9.6, Problem 76E

(a)

To determine

To calculate: The number of term required to approximate the sum of the series ∑k=0∞(−3)k1⋅3⋅5⋯(2k+1) with an error less than 0.0001.

(b)

To determine

To calculate: The approximatesum of the series ∑k=0∞(−3)k1⋅3⋅5⋯(2k+1) with an error less than 0.0001 using graphing utility.

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Chapter 9.6, Problem 75E

Chapter 9.6, Problem 77E

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Chapter 9 Solutions

Bundle: Calculus: Early Transcendental Functions, Loose-leaf Version, 6th + WebAssign Printed Access Card for Larson/Edwards' Calculus: Early Transcendental Functions, 6th Edition, Multi-Term

Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Prob. 6E Ch. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Prob. 9E Ch. 9.1 - Matching In Exercises 13-16, match the sequence...

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The number of term required to approximate the sum of the series ∑ k = 0 ∞ ( − 3 ) k 1 ⋅ 3 ⋅ 5 ⋯ ( 2 k + 1 ) with an error less than 0.0001 .

Solution Summary: The author calculates the number of terms required to approximate the sum of the series with an error less than 0.0001.

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To calculate: The number of term required to approximate the sum of the series ∑k=0∞(−3)k1⋅3⋅5⋯(2k+1) with an error less than 0.0001.

To calculate: The approximatesum of the series ∑k=0∞(−3)k1⋅3⋅5⋯(2k+1) with an error less than 0.0001 using graphing utility.

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Chapter 9 Solutions