009C Sample Midterm 1, Problem 5

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Find the radius of convergence and interval of convergence of the series.

(a)  

(b)  

Foundations:  
Ratio Test
        Let    be a series and  
        Then,

        If    the series is absolutely convergent.

        If    the series is divergent.

        If    the test is inconclusive.


Solution:

(a)

Step 1:  
We first 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 3:  
Now, we need to determine the interval of convergence.
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 4:  
First, let  
Then, the series becomes  
We note that
       
Therefore, the series diverges by the  th term test.
Hence, we do not include    in the interval.
Step 5:  
Now, let  
Then, the series becomes  
Since  
we have
       
Therefore, the series diverges by the  th term test.
Hence, we do not include    in the interval.
Step 6:  
The interval of convergence is  

(b)

Step 1:  
We first 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 3:  
Now, we need to determine the interval of convergence.
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 4:  
First, let  
Then, the series becomes  
This is an alternating series.
Let  .
First, we have
       
for all  
The sequence    is decreasing since
       
for all  
Also,
       
Therefore, this series converges by the Alternating Series Test
and we include    in our interval.
Step 5:  
Now, let  
Then, the series becomes  
First, we note that    for all  
Thus, we can use the Limit Comparison Test.
We compare this series with the series  
which is the harmonic series and divergent.
Now, we have

       

Since this limit is a finite number greater than zero,
        
diverges by the Limit Comparison Test.
Therefore, we do not include    in our interval.
Step 6:  
The interval of convergence is  


Final Answer:  
    (a)     The radius of convergence is    and the interval of convergence is  
    (b)     The radius of convergence is    and the interval of convergence is  

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