Forumula for the product of the first 'n' numbers?

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June 30th 2010, 09:00 AM
mfetch22

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:

$\sum_{i=1}^n i=\frac{(n)(n+1)}{2}$

I know that we can derive this formula by writing out the sum as

$S =1+2+ \cdot \cdot \cdot \cdot + (n-1) + (n)$

and doing some algebraic manipulation. But, what if we have the following:

$\prod_{i=1}^n i = (1)(2)(3) \cdot \cdot \cdot \cdot (n-2)(n-1)(n)$

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

$\sum_{i=1}^n i=\frac{(n)(n+1)}{2}$

such that we'd have something like the following

$\prod_{i=1}^n = f(n)$

[p.s. also, I realize that $n!$ 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 $(n)(n-1)(n-2)......(3)(2)(1)$ 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.]
June 30th 2010, 09:52 AM
Moo

Why do you want to find something more complicated ? You can view it like n*(n-1)! :D
June 30th 2010, 11:19 AM
roninpro

Quote:

Originally Posted by mfetch22

[p.s. also, I realize that $n!$ 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 $(n)(n-1)(n-2)......(3)(2)(1)$ 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.]

I'm a little bit concerned that your question is not well-posed. What do you mean by "precise formula"?
June 30th 2010, 12:54 PM
HallsofIvy

The reason why we define n! to be the product of the first n positive integers is precisely because there is no simple formula for that product!

I suspect that the reason there is a nice formula for the sum but not the product is that the ordering is defined by addition (1, 1+ 1, 1+ 1+ 1, ...) and multiplication just does not "play nicely" with addition.
June 30th 2010, 02:30 PM
theodds

It's worth mentioning Stirling's Approximation. The fact that people bother with this may help convince the OP that coming up with a "nice" formula for $n!$ is a lofty goal indeed.
June 30th 2010, 02:48 PM
chisigma

The product of the 'first n numbers' with n>0 is the value of the 'factorial function'...

$\displaystyle z!= \int_{0}^{\infty} t^{z}\ e^{-t}\ dt$ (1)

... when z=n (Rofl)...

Kind regards

$\chi$ $\sigma$