009C Sample Midterm 2, Problem 1 Detailed Solution

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Background Information:  
1. L'Hôpital's Rule, Part 2

        Let    and    be differentiable functions on the open interval    for some value   

        where    on    and    returns either    or   
2. The sum of a convergent geometric series is  
        where    is the ratio of the geometric series
        and    is the first term of the series.



Step 1:  


We then take the natural log of both sides to get
Step 2:  
We can interchange limits and continuous functions.
Therefore, we have


Now, this limit has the form  
Hence, we can use L'Hopital's Rule to calculate this limit.
Step 3:  
Now, we have


Step 4:  
Since    we know
Now, we have



Step 1:  
First, we not that this is a geometric series with  
this series converges.
Step 2:  
Now, we need to find the sum of this series.
The first term of the series is  
Hence, the sum of the series is


Final Answer:  

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