Science:Math Exam Resources/Courses/MATH105/April 2013/Question 06 (b)
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Question 06 (b) 

Find the interval of convergence of the following series: where for and 
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 

First, use the ratio test. 
Hint 2 

The second sentence about the telescoping sum seems to be obvious but it actually tells us something very important. Look at the partial sums and use this information to help evaluate the limit from the ratio test. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Let Ratio test gives Next, the partial summation from the convergent sum given to us in the problem statement tells us that Since these partial sums converge to , we know that and hence by relabeling Thus the reciprocal limit from the ratio test above diverges and so the limit in the ratio test diverges unless . Thus the interval of convergence is just the point . 