Explanation Given: The series is ∑ k = 1 ∞ ( − 1 ) k k 2 − 1 . Result used: (1) Let ∑ a k and ∑ b k are the series with positive terms and lim x → ∞ a n b n = L , where L is a finite number. (i) If 0 < L < ∞ , then either both series ∑ a k and ∑ b k converges or both diverges. (ii) If L = 0 and ∑ b k converges, then ∑ a k converges. (iii) If L = ∞ and ∑ b k diverges, then ∑ a k diverges” (2) The p -series ∑ k = 0 ∞ 1 k p converges if p > 1 Definition used: If ∑ a k converges absolutely if ∑ | a k | converges. Calculation: Consider the absolute value of the given series ∑ k = 1 ∞ 1 k 2 − 1 . k 2 > k 2 − 1 1 k 2 − 1 > 1 k 2 Consider the series ∑ k = 1 ∞ b k = ∑ k = 1 ∞ 1 k 2 , which must be greater than

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Author: Briggs

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Chapter 8, Problem 43RE

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Whether the series ∑k=1∞(−1)kk2−1 converges or diverges. State whether the convergence is conditional or absolute.

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Chapter 8, Problem 42RE

Chapter 8, Problem 44RE

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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 ∑ k = 1 ∞ ( − 1 ) k k 2 − 1 converges or diverges. State whether the convergence is conditional or absolute.

Solution Summary: The author states whether the series converges or diverges, and states if the convergence is conditional or absolute.

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Whether the series ∑k=1∞(−1)kk2−1 converges or diverges. State whether the convergence is conditional or absolute.

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