009C Sample Midterm 1, Problem 4 Detailed Solution

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Determine the convergence or divergence of the following series.

Be sure to justify your answers!

Background Information:  
Direct Comparison Test
        Let    and    be positive sequences where  
        for all    for some  
        1. If    converges, then    converges.
        2. If    diverges, then    diverges.


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:  
Also, we have    since
for all  
Therefore, the series    converges
by the Direct Comparison Test.

Final Answer:  
        converges (by the Direct Comparison Test)

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