Factorials

[StephenPrivitera]

We know that the sum of the first n natural numbers is n(n+1)/2

Can we express the product of the first n natural numbers without using the factorial symbol?
It is possible to write a factorial as a sum. Any idea what it would look like?

[Hurkyl]

To my knowledge, there isn't any convenient way.

You can do a "cheap" conversion from a product to a sum, though:

ln Πf(n) = Σln f(n)

So for factorials:

ln(n!) = Σln n
or
n! = eΣln n

[ahrkron]

Another way to write a factorial as a sum (which, I admit, sounds like cheating) is to use the Gamma function.

n! = Gamma(n+1) = Integral(tx-1e-t)dt

(the integral goes from zero to infinity)

Since an integral is the limit of a sum, it is kinda what you wanted.