Explanation Theorem used: The Comparison Test : Suppose that ∑ a n and ∑ b n are series with positive terms. a. If ∑ b n is convergent and a n ≤ b n for all n, then ∑ a n is also convergent. b. If ∑ b n is divergent and a n ≥ b n for all n, then ∑ a n is also divergent. Description: The proof is by using a counter example. Suppose that ∑ a n and ∑ b n are series with positive terms. If ∑ a n is convergent and b n ≤ a n for all n . Consider the series a n = ∑ 1 2 n 2 . It is observed that when n is large 2 n 2 behaves like n 2 . So a n behaves like b n = 1 n 2 if n ≥ 1 . Now compare ∑ a n with the test series ∑ b n

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Chapter 11.4, Problem 59E

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Whether the statement “Suppose that ∑an and ∑bn are series with positive terms. If ∑an is convergent and bn≥an for all n, then ∑bn is also divergent” is true or false; explain if it is true, or give an example if it is false.

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Chapter 11.4, Problem 58E

Chapter 11.4, Problem 60E

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Chapter 11 Solutions

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Whether the statement “Suppose that ∑ a n and ∑ b n are series with positive terms. If ∑ a n is convergent and b n ≥ a n for all n, then ∑ b n is also divergent” is true or false; explain if it is true, or give an example if it is false.

Solution Summary: The author explains whether the statement "Suppose that 'displaystyle' sum'a_n and

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Whether the statement “Suppose that ∑an and ∑bn are series with positive terms. If ∑an is convergent and bn≥an for all n, then ∑bn is also divergent” is true or false; explain if it is true, or give an example if it is false.

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Chapter 11 Solutions