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 . Calculation: The given geometric series is ∑ n = 1 ∞ ( − 1 ) n − 1 ( 1 2 ) n . Simplify the series ∑ n = 1 ∞ ( − 1 ) n − 1 ( 1 2 ) n as follows. ∑ n = 1 ∞ ( − 1 ) n − 1 ( 1 2 ) n = ∑ n = 1 ∞ ( − 1 ) n ( − 1 ) − 1 ( 1 2 ) n = ∑ n = 1 ∞ 1 − 1 ⋅ ( − 1 2 ) n = ∑ n = 1 ∞ ( − 1 ) ( − 1 2 ) n The series ∑ n = 1 ∞ ( − 1 ) ( − 1 2 ) n is a geometric series with a = − 1 and r = | − 1 2 | = 0

10th Edition

ISBN: 9781305041264

Author: Tan

Publisher: CENGAGE C

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 11, Problem 19RE

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To find: The sum of the geometric series ∑n=1∞(−1)n−1(12)n if it converges.

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Chapter 11, Problem 18RE

Chapter 11, Problem 20RE

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Chapter 11 Solutions

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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 sum of the geometric series ∑ n = 1 ∞ ( − 1 ) n − 1 ( 1 2 ) n if it converges.

Solution Summary: The author explains how to find the sum of the geometric series displaystyle undersetn=1overset

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To find: The sum of the geometric series ∑n=1∞(−1)n−1(12)n if it converges.

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Chapter 11 Solutions