Explanation Concept Used: A. The set of real numbers x where the series converges is the interval of convergence. B. Ratio Test: Let ∑ n = 1 ∞ a k be a series with nonzero terms. Let ρ = lim k → ∞ | a k + 1 a k | i. If ρ < 1 , then the series converges absolutely. ii. If ρ > 1 or ρ = ∞ , then the series diverges. iii. If ρ = 1 , the test does not provide any information. To find the Interval of convergence of a power series we start by testing the convergence of the series using various test like ratio test, root test etc. In this process we generally arrive at a condition on variable x for which the series may be convergent if this condition is satisfied for the test being used. These conditions help us figure out the interval of convergence. Given the power series ∑ k = 1 ∞ ( − 1 ) k + 1 x 2 k − 1 ( 2 k − 1 ) ! , we start by applying ratio test to find if this series is convergent or not. In this case ∑ n = 1 ∞ a k = ∑ k = 1 ∞ ( − 1 ) k + 1 x 2 k − 1 ( 2 k − 1 ) ! ⇒ a k = ( − 1 ) k + 1 x 2 k − 1 ( 2 k − 1 ) ! Therefore, ρ = lim k

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 8, Problem 67SP

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To calculate: the interval of convergence of the power series

x−x33!+x55!−x77!+…+(−1)k+1x2k−1(2k−1)!+⋯ .

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Chapter 8, Problem 66SP

Chapter 8, Problem 68SP

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Chapter 8 Solutions

Calculus

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the interval of convergence of the power series x − x 3 3 ! + x 5 5 ! − x 7 7 ! + … + ( − 1 ) k + 1 x 2 k − 1 ( 2 k − 1 ) ! + ⋯ .

Solution Summary: The author explains how to calculate the interval of convergence of a power series.

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To calculate: the interval of convergence of the power series

x−x33!+x55!−x77!+…+(−1)k+1x2k−1(2k−1)!+⋯ .

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the interval of convergence of the power series x − x 3 3 ! + x 5 5 ! − x 7 7 ! + … + ( − 1 ) k + 1 x 2 k − 1 ( 2 k − 1 ) ! + ⋯ .

Solution Summary: The author explains how to calculate the interval of convergence of a power series.

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To calculate: the interval of convergence of the power series x−x33!+x55!−x77!+…+(−1)k+1x2k−1(2k−1)!+⋯ .

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Chapter 8 Solutions

To calculate: the interval of convergence of the power series

x−x33!+x55!−x77!+…+(−1)k+1x2k−1(2k−1)!+⋯ .