Science:Math Exam Resources/Courses/MATH101/April 2016/Question 02 (c)
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Question 02 (c) 

For which values of the constant does the sum converge? L: It converges only when . M: It converges only when . N: It converges only when . P: It converges only when . Q: It converges for all . 
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 

Consider the integral test, and the conditions of using integral test 
Checking a solution serves two purposes: helping you if, after having used the hint, 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. First consider when . For , we have , which implies and . By the comparison test, we have
Therefore, for , the given series diverges.
In this case, we'll use the integral test. For but , we have
provided that , Here, the second equality is obtained from the substitution . Using ,
Then, by the integral test, the given series is convergent for ; ,
. Therefore, again by the integral test, the given series diverges when . To sum, the answer is L. 