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I'm doing a sequences problem where I have to write the first five terms of a sequence. It looks normal, but there is an exclamation mark on the denominator:

$$a_n = \frac{1}{(n + 1)!}$$

&

$$a_n = \frac{(-1)^nn}{n! + 1}$$

What does the exclamation mark mean, and how do I go about this differently with the exclamation mark?

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$n!$ is the factorial of $n$. (See the link.) –  David Mitra Apr 9 '12 at 0:01
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That’s the factorial sign: $n!=n(n-1)(n-2)\ldots\cdot2\cdot1$, and we define $0!=1$. Thus, $4!=4\cdot3\cdot2\cdot1=24$. –  Brian M. Scott Apr 9 '12 at 0:02
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It means that sequences are very exciting! –  alex.jordan Apr 9 '12 at 1:06
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1 Answer

up vote 5 down vote accepted

The factorial of a number is represented by the exclamation point (!). The factorial of a number $x$ is often described as the product of all positive integers less then or equal to $x$. For example:

$$4! = 4\cdot3\cdot2\cdot1 = 24$$

It is often also useful to describe $x!$ in a recursive relation:

$$x! = x(x-1)!$$

where $0! = 1$.

This method is often good because it helps explain why $0!=1$ (see also the "empty product" for more on this). Using the above explanation, we may find $4!$:

$$4! = 4(3)! = 4(3)(2)! = \cdots = 4(3)(2)(1)(0)! = 4(3)(2)(1)(1) = 24$$

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Awesome, thanks! –  Zolani13 Apr 9 '12 at 0:26
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