009C Sample Midterm 3, Problem 3 Detailed Solution

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Test if each the following series converges or diverges.

Give reasons and clearly state if you are using any standard test.



Background Information:  
1. Ratio Test
        Let    be a series and  

        If    the series is absolutely convergent.

        If    the series is divergent.

        If    the test is inconclusive.

2. If a series absolutely converges, then it also converges.
3. Limit Comparison Test
        Let    and    be positive sequences.
        If    where    is a positive real number,
        then    and    either both converge or both diverge.



Step 1:  
We begin by using the Ratio Test.
We have


Step 2:  
Since    the series is absolutely convergent by the Ratio Test.
Therefore, the series converges.


Step 1:  
First, we note that
for all  
This means that we can use a comparison test on this series.
Step 2:  
We want to compare the series in this problem with
This is a  -series with  
Hence,    converges.
Step 3:  
Now, we have
Therefore, the series
converges by the Limit Comparison Test.

Final Answer:  
    (a)     converges (by the Ratio Test)
    (b)     converges (by the Limit Comparison Test)

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