Explanation Result used: Alternating Series Test “The alternating series ∑ ( − 1 ) k + 1 a k converges if the terms of the series are non-increasing in magnitude ( 0 < a k + 1 ≤ a k , for k greater than some index N ) and lim k → ∞ a k = 0 , Otherwise the series diverges. Calculation: The given series ∑ k = 1 ∞ a k = ∑ k = 1 ∞ k ! k k . Here, the n th term is, a k = k ! k k . Thus, the ( n + 1 ) th term is, a k + 1 = ( k + 1 ) ! ( k + 1 ) k + 1 . Obtain the limit of | a k + 1 a k | . lim k → ∞ | a k + 1 a k | = lim k → ∞ | ( k + 1 ) ! ( k + 1 ) k + 1 k ! k k | = lim k → ∞ | ( k + 1 ) ! k k ( k + 1 ) k + 1 k ! | = lim k → ∞ | ( k + 1 ) k ! k k ( k + 1 ) k + <

Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

2nd Edition

ISBN: 9780321954329

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 8.6, Problem 26E

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Whether the series converges or not.

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Chapter 8.6, Problem 25E

Chapter 8.6, Problem 27E

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Chapter 8 Solutions

Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

Ch. 8.1 - Define sequence and give an example.Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Given the series k=1k, evaluate the first four...Ch. 8.1 - The terms of a sequence of partial sums are...Ch. 8.1 - Consider the infinite series k=11k. Evaluate the...Ch. 8.1 - Explicit formulas Write the first four terms of...Ch. 8.1 - Explicit formulas Write the first four terms of...

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Whether the series converges or not.

Solution Summary: The author explains that the series converges if the terms are non-increasing in magnitude.

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Whether the series converges or not.

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Chapter 8 Solutions