Science:Math Exam Resources/Courses/MATH103/April 2012/Question 02 (a)
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Question 02 (a) 

Short answer question. Simplify your answer to the best possible. Show in detail how you arrive at your answer (work will be considered for this problem). Use the integral comparison test to show that converges. 
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 comparison test, i.e. find an integrand g(x) such that
converges. 
Hint 2 

What can you say about 1/(2x1) on the interval on which the integration is considered? 
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. On the interval we have that 2x1 ≥ 1 and so for all x ≥ 1. Therefore by the integral comparison test. Since the latter integral converges (see lecture notes or direct calculation), the former integral also converges, by the integral comparison test. 