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Math Help - Empty Product Limit

  1. #1
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    Empty Product Limit

    Hi, I was wondering if for a product \prod_{i=1}^{n-1}f(n,i)=g(n) you can take the limit as n goes to 1 in the product, where g(1)\neq 1.

    In other words, can you explicitly take the limit \lim_{n\to 1}\prod_{i=1}^{n-1}f(n,i)? Assuming the relationship with g(n) holds for all n>0, the limit can't just be the empty product, hence my dilemma.

    Thanks!
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  2. #2
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    Quote Originally Posted by Texxy View Post
    Hi, I was wondering if for a product \prod_{i=1}^{n-1}f(n,i)=g(n) you can take the limit as n goes to 1 in the product, where g(1)\neq 1.

    In other words, can you explicitly take the limit \lim_{n\to 1}\prod_{i=1}^{n-1}f(n,i)? Assuming the relationship with g(n) holds for all n>0, the limit can't just be the empty product, hence my dilemma.

    Thanks!
    I am puzzled as to why there is a dilemma. According to your definition, g(1)= \Pi_{i=1}^0 f(1,i) which is itself an "empty product" and so is equal to 1. You can't just declare that g(1)\ne 1 unless you are defining g(1) separately, and not by that formula, in which case this function is not continuous and you can't expect a limit to give you the correct value.
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