What does ! mean in sequences?

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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?

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2 Answers

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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 $4!$:

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

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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.

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