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The following limit came up in the middle of solving a problem. Let $b$, $h$, and $k \geq 2$ be constants. If it helps simplifying the problem in any way, assume $b=2h$ and $k=2$.

I want to evaluate

$\underset{m \rightarrow +\infty}{\lim} \prod_{i=0}^m \left( \left(1-\frac{(1-2i)b}{4h}\frac{1}{m+1} \right)^{\left( \dfrac{4}{k^2}\dfrac{1}{m+1} - \dfrac{(1-2i)b}{hk^2}\dfrac{1}{(m+1)^2} \right)} \right).$

Numerical simulations show that this limit is a positive number less than 1. I would like to know how to evaluate this limit of product of infinite number of terms that approach 1.

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    @tatterdemalion, given that one can turn the product problem into a series problem (as I wrote out above), you could try using various series convergence tests to try to show the limit exists. I haven't tried it myself, though.2012-11-27

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