Science:Math Exam Resources/Courses/MATH103/April 2014/Question 11
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Question 11 

'This is a difficult question and will be marked very strictly. You should try this ONLY AFTER you finish all other problems. Suppose a sequence satisfies the recursive relation
where is a given positive constant. Suppose . Find all positive constants for which the series converges. Justify 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 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 

Do not use the ratio test (you will get a ratio of , so the test is indeterminate). 
Hint 2 

Use the divergence test. 
Hint 3 

When is a stable fixed point? When is it the only stable fixed point? 
Hint 4 

If is a fixed point of for which , then we can’t tell if it is stable or unstable. Make sure to check what happens in these special cases. 
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. By the divergence test, it is necessary (though perhaps not sufficient) that . Thus, the map must at least have a stable fixed point at . We thus begin by find the fixed points of . Solving yields or . To check stability, note that . Since , we require for to be stable; we will deal with the indeterminate case later. If , we use the ratio test to see that . We conclude by the ratio test that the series converges for and diverges for . In the case , we have . With , looking at the first few terms of the resulting sequence, we see that and so on. We thus see that . But then the series is the harmonic series, which diverges. Thus, the series converges for . 