MATH307 April 2009
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Question 07 (b)
Given the recurrence relation with the initial condition and .
(b) For what values of a and b will the sequence converge to a finite limit?
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Since every convergent sequence is Cauchy and every Cauchy sequence in converges, we can write the sequence as a Cauchy sequence to find the values of a and b for which the sequence converges to a finite limit. So for every there is a natural number N such that for every n and m integers greater than or equal to N, . Then , and so , that is, . Since can be arbitrarily large and we want this to be true for all , we propose that we need .
Suppose . Then , as required.
Now suppose there is some value of a such that but . Then let and take . Then . Since we can take n and m to be any integers such that they are greater than or equal to N, take and . Then since . This contradicts our assumption that there is a such that for any , .
So the sequence converges to a finite limit when b = a.