Science:Math Exam Resources/Courses/MATH105/April 2013/Question 02 (a)
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Question 02 (a) 

Find out whether the numerical series below converges or diverges. You should provide appropriate justification in order to receive credit. (a) 
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 

Try using the limit comparison test. You may also want to remind yourself of the pseries test. 
Hint 2 

Consider the dominant powers in the numerator and denominator of the terms in the series when determining a series to compare to when using the limit comparison test. 
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 

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Please rate my easiness! It's quick and helps everyone guide their studies. For this question, we apply the limit comparision test. Let We wish to define another series such that is positive and finite (i.e. ). By comparing the dominant powers in the numerator (1/3) and denominator (2), we are inspired (1/32 = 5/3) to define as The terms , are positive for all . By the limit comparison test, if is positive and finite, then both series converge or both diverge. We can see that will converge by the pseries test, since the power on in the denominator, , is greater than . So if is finite, then converges. So is positive and finite. Therefore, by the limit comparison test, the sum converges. 