Science:Math Exam Resources/Courses/MATH105/April 2012/Question 02 (a)
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Question 02 (a) 

This problem contains three numerical series. For each of them, ﬁnd out whether it converges or diverges. You should provide appropriate justiﬁcation in order to receive credit. 
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 

Use the integral test. 
Hint 2 

The resulting integral can be solved using substitution. 
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. When we look at this series, all its terms are positive, and they are clearly decreasing: is getting smaller and smaller, as is . The integral test can be applied here. We know that The integral I is improper (the infinite range of integration), but it can be evaluated with substitution. Set then , and therefore . From the above computation, I converges, and so does the original series. 