# Sum of the first n natural numbers is n(n+1)/2

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?

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## Answers and Replies

Hurkyl
Staff Emeritus
Gold Member
To my knowledge, there isn't any convenient way.

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

ln &Pi;f(n) = &Sigma;ln f(n)

So for factorials:

ln(n!) = &Sigma;ln n
or
n! = e&Sigma;ln n

ahrkron
Staff Emeritus
Gold Member
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.