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

Question

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

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

The domain is $n>1$

Answer 1

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.

Answer 2

$n!$ 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 $$\prod_{k=1}^{n}{k}$$