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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:

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

&

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

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

  • 1
    $n!$ is the [factorial](http://en.wikipedia.org/wiki/Factorial) of $n$. (See the link.)2012-04-09
  • 1
    That’s the [factorial](http://en.wikipedia.org/wiki/Factorial) sign: $n!=n(n-1)(n-2)\ldots\cdot2\cdot1$, and we define $0!=1$. Thus, $4!=4\cdot3\cdot2\cdot1=24$.2012-04-09
  • 22
    It means that sequences are very exciting!2012-04-09

2 Answers 2

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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 $4!$:

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

  • 0
    Awesome, thanks!2012-04-09
  • 0
    For the record: $0!=1$. ${}\qquad{}$2015-12-23
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If the number is 4! then you count down so it would be $4\times 3 \times 2\times 1= 24$. Another thing is that if the number is 7! you would count down until you get to two. So it would be $7\times 6\times 5\times 3\times 2 =5040$ 7 x 6 x 5 x 4 x 3 x 2 I just put in the 4 that was not listed although the answer is still 5040.

  • 0
    It might be confusing to someone who has never seen factorials before if you count down to 1 in one example and 2 in another example (even though the result is the same).2014-05-19
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    @Hockeylover It's actually defined recursively and one "counts down" to $0! :=1$.2015-01-07