I know that the Euler-Maclaurin formula establishes a relationship between sums and integrals, but is there some sort of formula that establishes a relationship between products and integrals? I don't mean the product rule, for products of functions. I mean a way to convert the sort of product you use the Π notation for into an integral. Specifically, a product of all the values of a function, from f(1) to f(n), into an integral.
Sorry if this is a silly question; I'd still like to know the answer. Thanks very much.
Assuming $f(i) > 0, \forall i$, use can use the identity $$\prod_{i=1}^n f(i) = \exp \left[\sum_{i=1}^n \log f(i)\right]$$ and then employ Euler-Maclaurin for the sum on the right hand side.