Explanation Given : A sequence { a n } Calculation : Step 1: Let { a n } be a sequence such that lim n → ∞ | a n | exists and is nonzero. Suppose lim n → ∞ | a n | exists and let L= lim n → ∞ | a n | . Note that L cannot be zero for then lim n → ∞ | a n | also be zero. Step 2: Now choose ε < | L | . Then there exists an integer M such that | a n − L | < ε or L − ε < a n < L + ε for all n>M. Step 3: If L<0, then − 2 L < a n < 0 whereas if L>0, then 0 < a n < 2 L . This implies that a n does not change for n > M . Step 4: Now suppose that there exists an integer M such that a n does not change for n > M ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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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 11.1, Problem 86E

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The mathematical proof of the statement stating that $\lim_{n \to \infty} | a_{n} |$ exists, if and only if there exists an integer M such that the sign of $a_{n}$ does not change for $n > M$ .

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Chapter 11.1, Problem 85E

Chapter 11.1, Problem 87E

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CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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The mathematical proof of the statement stating that lim n → ∞ | a n | exists, if and only if there exists an integer M such that the sign of a n does not change for n > M .

Solution Summary: The author explains the mathematical proof of the statement stating that lim n exists, if and only when there exists an integer M such that the sign of a

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The mathematical proof of the statement stating that $\lim_{n \to \infty} | a_{n} |$ exists, if and only if there exists an integer M such that the sign of $a_{n}$ does not change for $n > M$ .

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Chapter 11 Solutions

The mathematical proof of the statement stating that lim n → ∞ | a n | exists, if and only if there exists an integer M such that the sign of a n does not change for n &gt; M .

Solution Summary: The author explains the mathematical proof of the statement stating that lim n exists, if and only when there exists an integer M such that the sign of a

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The mathematical proof of the statement stating that limn→∞|an| exists, if and only if there exists an integer M such that the sign of an does not change for n>M.

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Chapter 11 Solutions

The mathematical proof of the statement stating that lim n → ∞ | a n | exists, if and only if there exists an integer M such that the sign of a n does not change for n > M .

The mathematical proof of the statement stating that $\lim_{n \to \infty} | a_{n} |$ exists, if and only if there exists an integer M such that the sign of $a_{n}$ does not change for $n > M$ .