Explanation Theorem used: Suppose the power series is given by ∑ n = 0 ∞ a n ( x − a ) n and let R = lim n → ∞ | a n a n + 1 | . a. If R = 0 , then the series converges only for x = a . b. If 0 < R < ∞ , the series converges for x in the interval ( a − R , a + R ) and diverges for x outside the interval [ a − R , a + R ] . c. If R = ∞ , the series converges for all x. Calculation: The given power series is ∑ n = 0 ∞ x n n 2 + 2 . Here, a n = 1 n 2 + 2 . By using the above mentioned theorem, obtain the radius of convergence of the power series ∑ n = 0 ∞ x n n 2 + 2 is R = lim n → ∞ | 1 n 2 + 2 1 ( n + 1 ) 2 + 2 | . Solve R = lim n → ∞ | 1 n 2 + 2 1 ( n + 1 ) 2 + 2 | as follows. R = lim n → ∞ | 1 n 2 + 2 1 ( n + 1 ) 2 + 2 | = lim n → ∞ ( ( n + 1 ) 2 + 2 n 2 + 2 )

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 11, Problem 33RE

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To find: The radius of convergence and the interval of convergence of the power series ∑n=0∞xnn2+2.

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Chapter 11, Problem 32RE

Chapter 11, Problem 34RE

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Chapter 11 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

Ch. 11.1 - Prob. 1CQ Ch. 11.1 - Prob. 2CQ Ch. 11.1 - Prob. 1E Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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The radius of convergence and the interval of convergence of the power series ∑ n = 0 ∞ x n n 2 + 2 .

Solution Summary: The author calculates the radius and interval of convergence of the given power series using the above-mentioned theorem.

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To find: The radius of convergence and the interval of convergence of the power series ∑n=0∞xnn2+2.

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Chapter 11 Solutions