Science:Math Exam Resources/Courses/MATH103/April 2016/Question 07 (b) (ii)
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Question 07 (b) (ii) 

Consider the power series: . (ii) Find all values of such that the power series 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 

When consider the pseries test. 
Hint 2 

When apply the alternating series test. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From part (i), we already know that the given series converges when and diverges when Therefore, it is enough to consider its convergence/divergence when i.e., when
where the last equality follows from the pseries test (with p = 1/2).
Then, since is a decreasing sequence converging to 0, by the alternating series test, the series converges.
