Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (c)
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Question 05 (c) 

Determine whether the series converges or diverges. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Apply the comparison and pseries tests. 
Hint 2 

Use the fact that for all m to help you find a series to compare the given one to. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We will apply the comparison test to show that the series diverges. The comparison test states that if and are sequences with for , then if diverges, the series diverges as well. Since for all , . Furthermore, for all , so As diverges by the pseries test (here, ), then by the comparison test, the series diverges as well. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We use the limit comparison test to show that the series diverges. This test says that if are sequences with , and if (and the limit exists), then the series and either both converge or both diverge. Applying the test with and , we observe that which clearly lies between and . As diverges (by the pseries test with ), the series also diverges. 