Science:Math Exam Resources/Courses/MATH101/April 2016/Question 11 (a)
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Question 11 (a) 

Both parts of this question concern the series . (a) Show that the series converges absolutely. 
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 either the Integral test or 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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Absolutly convergent series is when converges, so we need to show that converges. We can apply the integral test by setting , first we should check if the conditions of the integral test hold: First, is certainly positive. Then we find
which is negative for . Therefore is decreasing for , and the Integral Test applies. Note that the substitution yields
Therefore,
Since the integral is convergent, converges absolutely. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Alternatively, we can use the Ratio Test. We calculate
and therefore the series converges absolutely. 