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

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

Given the appearance of factorial terms in the series, you might try applying the ratio 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. We apply the ratio test for this problem. Let By the ratio test, if , then the series converges. If , the series diverges. If , then no conclusion can be drawn. For , all are positive so we can remove the absolute value signs in when evaluating it. Using the fact that and we can simplify the fractions with factorial terms, giving us the following: Dividing both the numerator and denominator by gives: Since , this series diverges by the ratio test. 