Forumula for the product of the first 'n' numbers?
I know that we are able to derive a formula for the sum of the first 'n' positive natural numbers, that sum being:
$\displaystyle \sum_{i=1}^n i=\frac{(n)(n+1)}{2}$
I know that we can derive this formula by writing out the sum as
$\displaystyle S =1+2+ \cdot \cdot \cdot \cdot + (n-1) + (n)$
and doing some algebraic manipulation. But, what if we have the following:
$\displaystyle \prod_{i=1}^n i = (1)(2)(3) \cdot \cdot \cdot \cdot (n-2)(n-1)(n)$
Is there a way in which we can manipulate the product expansion (simmilar to the way in which we can manipulate the summation expansion) which will cause us to arrive at a direct formula for the product of the first 'n' positive integers simmilar to
$\displaystyle \sum_{i=1}^n i=\frac{(n)(n+1)}{2}$
such that we'd have something like the following
$\displaystyle \prod_{i=1}^n = f(n)$
[p.s. also, I realize that $\displaystyle n!$ would denote the function I am refering to, but that sort if "notation" is not what I am refering to here. I'm asking weather there exists an explicit forumula for the product which does not have to be written in the form $\displaystyle (n)(n-1)(n-2)......(3)(2)(1)$ or any form that includes the ".....", but rather a precise formula simmilar to the one used for the sum of the first "n" positive integers.]
