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