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Math Help - Very Hard Limit

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    Very Hard Limit

    Verify numerically that:

    limit as n->infinity of ((e^n)*(n!))/(n^(n+1/2))=sqrt(2*pi)

    What do they mean by numerically? I know I could do L'H rule, but then how would I find the derivative of n!?
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  2. #2
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    Quote Originally Posted by twittytwitter View Post
    Verify numerically that:

    limit as n->infinity of ((e^n)*(n!))/(n^(n+1/2))=sqrt(2*pi)

    What do they mean by numerically? I know I could do L'H rule, but then how would I find the derivative of n!?
    You're probably expected to substitute larger and larger values of n and note the sequence of resulting values ....
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    Ok thanks, so there's no algebraic way I could show this that is not too complicated?
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    Quote Originally Posted by twittytwitter View Post
    Ok thanks, so there's no algebraic way I could show this that is not too complicated?
    There is. But:

    Quote Originally Posted by twittytwitter View Post
    Verify numerically that:

    limit as n->infinity of ((e^n)*(n!))/(n^(n+1/2))=sqrt(2*pi)

    What do they mean by numerically? I know I could do L'H rule, but then how would I find the derivative of n!?
    which means that an algebraic way is probably beyond your present capability.
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    Quote Originally Posted by twittytwitter View Post
    Ok thanks, so there's no algebraic way I could show this that is not too complicated?
    Check out Stirling's approximation - Wikipedia, the free encyclopedia

    What you have is a somewhat inverted version of this approximation. There are many ways to show this, the two that I know involve approximating that Gamma integral for large argument.
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