Explanation Concept Used: Ratio Test: Let ∑ k = 1 ∞ a k be a series with nonzero terms. Let ρ = lim k → ∞ | a k + 1 a k | i. If ρ < 1 , then the seriesconverges absolutely. ii. If ρ > 1 or ρ = ∞ , then the seriesdiverges. iii. If ρ = 1 , the test does not provide any information. It is given that { a n } is a sequence such that | a n | < A n for some positive number A and all positive integers n . Observe that | a n | < A n ⇒ | a n + 1 a n | < A n + 1 A n = A ⇒ | a n + 1 a n | < A According to absolute convergenceTest, for a given series ∑ a k , if the series ∑ | a k | converges then the series ∑ a k converges absolutely. For absolute convergence of the series ∑ a k x k we need the series ∑ | a k x k | to converge

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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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 8.6, Problem 54PS

To determine

To show:that the series ∑akxk converges absolutely for |x|≤1A ..

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Chapter 8.6, Problem 53PS

Chapter 8.6, Problem 55PS

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Chapter 8 Solutions

Calculus

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that the series ∑ a k x k converges absolutely for | x | ≤ 1 A ..

Solution Summary: The author explains that the series displaystyle'sum' converges absolutely for a series with nonzero terms.

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To show:that the series ∑akxk converges absolutely for |x|≤1A ..

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Chapter 8 Solutions