Science:Math Exam Resources/Courses/MATH101/April 2017/Question 01 (c)
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Question 01 (c) 

For which real numbers does the power series converge? 
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 ratio test, the pseries test, and the alternating series 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, apply the ratio test for the given series . Since
by the ratio test, the series converges for satisfying . (i.e., ) and diverges for satisfying (i.e., ),
.
. Since decreasing and converges to as goes to infinity, by the alternating series test, it converges. To summarize, the given series converges when . 