Science:Math Exam Resources/Courses/MATH101/April 2011/Question 08
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Question 08 

Find (a) the radius of convergence, and (b) the interval of convergence of the following power series, carefully justifying your answer: 
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 

a.) Ratio Test b.) Check end points 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To find the radius of convergence, we use the ratio test. Let To see that note that Since logarithm is continuous on its domain, the numerator satisfies In the denominator, as . We conclude that ; that is, as By the ratio test, it follows that the series is absolutely convergent if and only if Therefore, the radius of convergence is equal to 1. In order to determine the interval of convergence, it remains to determine whether the series is convergent at the endpoints, At the series is equal to By comparison with this series is divergent. (To compare, note that ) At the series is equal to By the alternating series test, this converges. Hence, the interval of convergence is . 