009C Sample Midterm 2, Problem 3 Detailed Solution

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Determine convergence or divergence:



Background Information:  
1. Alternating Series Test
        Let    be a positive, decreasing sequence where  
        Then,    and  
2. Ratio Test
        Let    be a series and  

        If    the series is absolutely convergent.

        If    the series is divergent.

        If    the test is inconclusive.

3. If a series absolutely converges, then it also converges.



Step 1:  
First, we have
Step 2:  
We notice that the series is alternating.
First, we have
for all  
The sequence    is decreasing since
for all  
Therefore, the series    converges by the Alternating Series Test.


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


Step 2:  
Now, we need to calculate  
Then, taking the natural log of both sides, we get


since we can interchange limits and continuous functions.
Now, this limit has the form  
Hence, we can use L'Hopital's Rule to calculate this limit.
Step 3:  
Now, we have


Step 4:  
Since    we know
Now, we have
Since    the series is absolutely convergent by the Ratio Test.
Therefore, the series converges.

Final Answer:  
    (a)     converges (by the Alternating Series Test)
    (b)     converges (by the Ratio Test)

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