009C Sample Midterm 1, Problem 1

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Does the following sequence converge or diverge?

If the sequence converges, also find the limit of the sequence.

Be sure to jusify your answers!


Foundations:  
L'Hôpital's Rule, Part 2

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

        where    on    and    returns either    or   
       Then,  


Solution:

Step 1:  
First, notice that
       
and
       
Therefore, the limit has the form  
which means that we can use L'Hopital's Rule to calculate this limit.
Step 2:  
First, switch to the variable     so that we have functions and
can take derivatives. Thus, using L'Hopital's Rule, we have
       


Final Answer:  
        The sequence converges. The limit of the sequence is  

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