1. ## Empty Product Limit

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

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

Thanks!

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

In other words, can you explicitly take the limit $\displaystyle \lim_{n\to 1}\prod_{i=1}^{n-1}f(n,i)$? Assuming the relationship with $\displaystyle g(n)$ 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, $\displaystyle g(1)= \Pi_{i=1}^0 f(1,i)$ which is itself an "empty product" and so is equal to 1. You can't just declare that $\displaystyle g(1)\ne 1$ 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.