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Math Help - Limit (~ is Landau's symbol in this Qs)

  1. #1
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    Smile Limit (~ is Landau's symbol in this Qs)

    Hi everyone:
    I got a question which is about limit, after thinking still don't know how to starts it.
    please help!!!!

    Qrove that (n+a)!/(n+b)! ~ n^(a-b) as n goes to +ive infinite (for fixed a,b>0)

    and the Qs provide some formula to use
    Euler's limit: '[1+(x/n)]^n' → 'e^x' as n goes to infinite
    Stirling's formula: n! ~ [root(2πn)](n/e)^n

    Thanks very much.
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  2. #2
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    Quote Originally Posted by wsun View Post
    Hi everyone:
    I got a question which is about limit, after thinking still don't know how to starts it.
    please help!!!!

    Qrove that (n+a)!/(n+b)! ~ n^(a-b) as n goes to +ive infinite (for fixed a,b>0)

    and the Qs provide some formula to use
    Euler's limit: '[1+(x/n)]^n' → 'e^x' as n goes to infinite
    Stirling's formula: n! ~ [root(2πn)](n/e)^n

    Thanks very much.
    Start by using Stirling's formula to replace the factorials in (n+a)!/(n+b)!

    (you might want to consider why what you know means:

    \frac{(n+a)!}{(n+b)!} \sim \frac{\sqrt{2\pi (n+a)}((n+a)/e)^{n+a}}{\sqrt{2\pi (n+b)}((n+b)/e)^{n+b}} \sim \frac{((n+a)/e)^{n+a}}{((n+b)/e)^{n+b}}

    Then use the other limit)

    CB
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  3. #3
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    I got it.
    Thanks
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