009B Sample Final 3, Problem 7

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Does the following integral converge or diverge? Prove your answer!

Foundations:  
Direct Comparison Test for Improper Integrals
        Let    and    be continuous on  
        where    for all    in  
       1.  If    converges, then    converges.
       2.  If    diverges, then    diverges.


Solution:

Step 1:  
We use the Direct Comparison Test for Improper Integrals.
For all    in  
       
Also,
         and  
are continuous on  
Step 2:  
Now, we have
       
Since    converges,
       
converges by the Direct Comparison Test for Improper Integrals.


Final Answer:  
       converges (by the Direct Comparison Test for Improper Integrals)

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