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This question already has an answer here:

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

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

The domain is $n>1$

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

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

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

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