009C Sample Midterm 2, Problem 2 Detailed Solution

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Determine convergence or divergence:

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 the harmonic series (or  -series with   )
Hence,    diverges.
Step 3:  
Also, we have    since
for all  
Therefore, the series    diverges
by the Direct Comparison Test.

Final Answer:  
        diverges (by the Direct Comparison Test)

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