MATH101 April 2014
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Question 01 (g)
Short Problem. Show your work. No credit will be given for the answer without the correct accompanying work.
Find the radius of convergence for the power series .
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.
Try using the ratio test.
The ratio test applies to series of the form and states that if then:
- If , is absolutely convergent
- If , is divergent
- If , the ratio test is inconclusive and so we must use another test to determine convergence
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Applying the ratio test with
we find that
The radius of convergence is the bound on values that allows the series to converge. Using the ratio test, the series converges if the ratio tends to a number smaller than 1. Hence the series converges for by the ratio test, and so its radius of convergence is 1.
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