Science:Math Exam Resources/Courses/MATH105/April 2016/Question 04 (b)
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Question 04 (b) 

Let , , and be the radius of convergence of the power series . Find 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 

To find the radius of convergence, consider the ratio test. 
Hint 2 

If a function has of the power series form and a point is in the interval of convergence, then the derivative of at is

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. We can rewrite the function in the power series form as follows:
In order to find the radius of convergence for a power series, we first apply the ratio test. So,
Then, the power series converges when , (i.e., ), while it diverges when , (i.e., ). Therefore, the radius of convergence is
We now compute
Note that the last series is a geometric series with the first term and the common ratio To sum up, the answers are . 