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Consider the series
(a) Test if the series converges absolutely. Give reasons for your answer.
(b) Test if the series converges conditionally. Give reasons for your answer.
Foundations:
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1. A series is absolutely convergent if
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the series converges.
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2. A series is conditionally convergent if
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the series diverges and the series converges.
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Solution:
(a)
Step 1:
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First, we take the absolute value of the terms in the original series.
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Let
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Therefore,
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Step 2:
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This series is a -series with
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Therefore, it diverges.
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Hence, the series
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is not absolutely convergent.
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(b)
Step 1:
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For
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we notice that this series is alternating.
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Let
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First, we have
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for all
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The sequence is decreasing since
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for all
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Also,
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Therefore, the series converges
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by the Alternating Series Test.
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Step 2:
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Since the series is not absolutely convergent but convergent,
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this series is conditionally convergent.
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Final Answer:
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(a) not absolutely convergent (by the -series test)
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(b) conditionally convergent (by the Alternating Series Test)
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