009C Sample Final 3, Problem 3

From Math Wiki
Jump to navigation Jump to search

Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

Limit Comparison Test
        Let    and    be positive sequences.
        If    where    is a positive real number,
        then    and    either both converge or both diverge.


Step 1:  
First, we note that
for all  
This means that we can use a comparison test on this series.
Step 2:  
We want to compare the series in this problem with
This is a  -series with  
Hence,    converges
Step 3:  
Now, we have
Therefore, the series
converges by the Limit Comparison Test.

Final Answer:  
        converges (by the Limit Comparison Test)

Return to Sample Exam