# How can I write $n!$ using $\sum$? [duplicate]

How can I write $n!$ using $\sum$?

Should I write $\sum\limits_{k=2}^n k$?

The domain is $n>1$

## marked as duplicate by Clarinetist, Kevin Long, Andrés E. Caicedo, Math Lover, ShaileshMay 11 '18 at 0:06

• do you know what $n!$ means? – JustDroppedIn May 10 '18 at 17:22
• Maybe $n! = \exp\left(\sum_{k=1}^n \log k \right)$ – angryavian May 10 '18 at 17:22
• @JustDroppedIn It's the factorial – user557276 May 10 '18 at 17:22
• Are you sure you don't mean $\Pi$ for product? – Kevin Long May 10 '18 at 17:23
• May be $$n!=\sum_{k=1}^{n}(n-1)!$$ – Bumblebee May 10 '18 at 17:51

Note that for non-negative integer $n$, $$n! = 1 \times 2 \times \ldots \times n = \prod_{k=1}^n k.$$ If you want to write $n!$ using a sum-like expression, note that $$\ln(n!) = \ln\left(\prod_{k=1}^n k \right) = \sum_{k=1}^n \ln k,$$ so $$n! = \exp\left(\sum_{k=1}^n \ln k\right),$$ but not sure this is what you are looking for.
$n!$ involves the multiplication of subsequent terms, rather than an addition, and therefore it is a product rather than a sum. Therefore, we would typically use product notation, signified by $\prod$ as opposed to $\sum$.
The best expression is $$\prod_{k=1}^{n}{k}$$