009C Sample Final 2, Problem 4

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(a) Find the radius of convergence for the power series

(b) Find the interval of convergence of the above series.

Ratio Test
        Let    be a series and  

        If    the series is absolutely convergent.

        If    the series is divergent.

        If    the test is inconclusive.



Step 1:  
We use the Ratio Test to determine the radius of convergence.
We have


Step 2:  
The Ratio Test tells us this series is absolutely convergent if  
Hence, the Radius of Convergence of this series is  


Step 1:  
First, note that    corresponds to the interval  
To obtain the interval of convergence, we need to test the endpoints of this interval
for convergence since the Ratio Test is inconclusive when  
Step 2:  
First, let  
Then, the series becomes  
This is an alternating series.
Let  .
The sequence    is decreasing since
for all  
Therefore, this series converges by the Alternating Series Test
and we include    in our interval.
Step 3:  
Now, let  
Then, the series becomes  
This is a  -series with    Hence, the series diverges.
Therefore, we do not include    in our interval.
Step 4:  
The interval of convergence is  

Final Answer:  
    (a)     The radius of convergence is  

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