Explanation Theorem used: If | r | < 1 , then the geometric series ∑ n = 0 ∞ a r n = a + a r + a r 2 + ⋯ converges and it sum is a 1 − r ; that is, ∑ n = 0 ∞ a r n = a + a r + a r 2 + ⋯ = a 1 − r . The series diverges if | r | ≥ 1 . Properties of Infinite Series: If ∑ n = 1 ∞ a n and ∑ n = 1 ∞ b n are convergent infinite series and c is a constant, then 1. ∑ n = 1 ∞ c a n = c ∑ n = 1 ∞ a n 2. ∑ n = 1 ∞ ( a n ± b n ) = ∑ n = 1 ∞ a n ± ∑ n = 1 ∞ b n Calculation: The given series is ∑ n = 1 ∞ [ ( e π ) n + ( π e 2 ) n ] . The series ∑ n = 1 ∞ [ ( e π ) n + ( π e 2 ) n ] can be rewritten as follows. ∑ n = 1 ∞ [ ( e π ) n + ( π e 2 ) n ] = ∑ n = 1 ∞ ( e π ) n + ∑ n = 1 ∞ ( π e 2 ) n The series ∑ n = 1 ∞ ( e π ) n is a geometric series with a = 1 and r = | e π | = 0.87 < 1 . The series ∑ n = 1 ∞ ( π e 2 ) n is a geometric series with a = 1 and r = | π e 2 | = 0.43 < 1 . From the above mentioned properties, it is concluded that the series ∑ n = 1 ∞ ( e π ) n + ∑ n = 1 ∞ ( π e 2 ) n is convergent. Now find its sum. Here, the summation begins with n = 1

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 11.3, Problem 25E

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Whether the series ∑n=1∞[(eπ)n+(πe2)n] converges or diverges; find its sum if it converges.

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Chapter 11.3, Problem 24E

Chapter 11.3, Problem 26E

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Chapter 11 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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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Whether the series ∑ n = 1 ∞ [ ( e π ) n + ( π e 2 ) n ] converges or diverges; find its sum if it converges.

Solution Summary: The author analyzes whether the geometric series displaystyle 'underset' is converged or diverges.

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Whether the series ∑n=1∞[(eπ)n+(πe2)n] converges or diverges; find its sum if it converges.

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Chapter 11 Solutions