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

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Apply the comparison and pseries tests. 
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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 converges. The comparison test states that if and are two sequences with for , then if converges, the series converges as well. Since , and . Hence . Furthermore, since for all , we have . Therefore, Since converges by the pseries test (here, ), by comparison the series also converges. 
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 converges. 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 converges (by the pseries test with ), the series also converges. 