009C Sample Final 2, Problem 1

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Test if the following sequences converge or diverge. Also find the limit of each convergent sequence.



L'Hopital's Rule

        Suppose that     and     are both zero or both  

       If     is finite or  




Step 1:  
First, we notice that    has the form  
So, we can use L'Hopital's Rule. To begin, we write
Step 2:  
Now, using L'Hopital's rule, we get


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

Final Answer:  

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