009C Sample Final 1, Problem 4

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Find the interval of convergence of the following series.

1. Ratio Test Let be a series and Then,
If the series is absolutely convergent.
If the series is divergent.
If the test is inconclusive.
2. After you find the radius of convergence, you need to check the endpoints of your interval
for convergence since the Ratio Test is inconclusive when


Step 1:  
We proceed using the ratio test to find the interval of convergence. So, we have
Step 2:  
So, we have Hence, our interval is But, we still need to check the endpoints of this interval
to see if they are included in the interval of convergence.
Step 3:  
First, we let Then, our series becomes
Since we have Thus, is decreasing.
So, converges by the Alternating Series Test.
Step 4:  
Now, we let Then, our series becomes
This is a convergent series by the p-test.
Step 5:  
Thus, the interval of convergence for this series is
Final Answer:  

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