Proving the Limit: (n+a)!/(n+b)! as n Goes to Infinity

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The discussion focuses on proving the limit of the expression (n+a)!/(n+b)! as n approaches infinity, concluding that it approximates to n^(a-b). Key tools mentioned for solving this problem include Euler's limit and Stirling's formula. Participants emphasize that advanced techniques are unnecessary, as the dominant factors are those containing n, while constants a and b become negligible in the limit. The polynomial expansion method is also suggested for a clearer understanding of the behavior of the expression as n increases.

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yanjt
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Hi,I have no idea on how to begin with this question.The question is:

Prove that (n+a)!/(n+b)! ~ na-b as n goes to infinity.There are clue given that we can use Euler's limit and Stirling's formula to solve this question.Can you please give me some hints on how to start with this question?

Thanks!
 
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I don't think any advanced machinery is necessary here. For any n, the term n!/n! = 1 can be ignored so we have left only a factors containing n in the numerator and b factors containing n in the denominator. You can then either note the polynomial expansion of the top and bottom, or note that a and b are constants, and thus are negligible as n increases without bound.
 

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