009C Sample Final 1, Problem 3

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Determine whether the following series converges or diverges.

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


Step 1:  
We proceed using the ratio test.
We have
Step 2:  
Now, we continue to calculate the limit from Step 1. We have
Step 3:  
Now, we need to calculate
First, we write the limit as
Now, we use L'Hopital's Rule to get
Step 4:  
We go back to Step 2 and use the limit we calculated in Step 3.
So, we have
Thus, the series absolutely converges by the Ratio Test.
Since the series absolutely converges, the series also converges.
Final Answer:  
   The series converges.

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