Explanation Theorem used: Test for divergence : If lim n → ∞ a n does not exist or lim n → ∞ a n ≠ 0 , then ∑ n = 1 ∞ a n diverges. The Integral Test : Suppose that f is a continuous, positive, and decreasing function on [ 1 , ∞ ] . If f ( n ) = a n for all n ≥ 1 , then ∑ n − 1 ∞ a n and ∫ 1 ∞ f ( x ) d x either both converge or both diverge. Description: The following counter example disprove the given statement as follows. Consider the series ∑ n = 1 ∞ 1 n . It is known that, the series ∑ n = 1 ∞ 1 n is divergent

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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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 11.4, Problem 57E

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Whether the statement “If ∑n=1∞an diverges, then limn→∞an≠0” is true or false.

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Chapter 11.4, Problem 56E

Chapter 11.4, Problem 58E

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Chapter 11 Solutions

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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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Whether the statement “If ∑ n = 1 ∞ a n diverges, then lim n → ∞ a n ≠ 0 ” is true or false.

Solution Summary: The author analyzes whether the statement is true or false. If f is a continuous, positive, and decreasing function on left[1,inftyright], the counter example disproves

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Whether the statement “If ∑n=1∞an diverges, then limn→∞an≠0” is true or false.

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Chapter 11 Solutions