Explanation Given: The series is ∑ k = 1 ∞ 1 k k + 1 . Result used: (1) “Let ∑ a k and ∑ b k are the series with positive terms and lim n → ∞ 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 = 1 ∞ 1 k p is convergent if p > 1 and diverges if p ≤ 1 . Calculation: Consider the series ∑ k = 1 ∞ 1 k k + 1 . k < k k + 1 1 k k + 1 < 1 k Consider the series ∑ k = 1 ∞ b k = ∑ k = 1 ∞ 1 k , which must be smaller than ∑ k = 1 ∞ a k = ∑ k = 1 ∞ 1 k k + 1

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

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Chapter 8, Problem 28RE

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Whether the series ∑ k = 1 ∞ 1 k k + 1 converges or diverges.

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Whether the series ∑k=1∞1kk+1 converges or diverges.

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