009C Sample Final 3, Problem 2
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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 |
the series converges. |
2. A series is conditionally convergent if |
the series diverges and the series converges. |
Solution:
(a)
Step 1: |
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First, we take the absolute value of the terms in the original series. |
Let |
Therefore, |
Step 2: |
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This series is a -series with |
Therefore, it diverges. |
Hence, the series |
is not absolutely convergent. |
(b)
Step 1: |
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For |
we notice that this series is alternating. |
Let |
First, we have |
for all |
The sequence is decreasing since |
for all |
Also, |
Therefore, the series converges |
by the Alternating Series Test. |
Step 2: |
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Since the series is not absolutely convergent but convergent, |
this series is conditionally convergent. |
Final Answer: |
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(a) not absolutely convergent (by the -series test) |
(b) conditionally convergent (by the Alternating Series Test) |