What does '!' mean when used in mathematics?

i.e: 5! = 5.4.3.2.1 = 120 

so => n! = n(n - 1)(n - 2).... 

by definition: 0! = 1 
so if n = 0, then n! = 1 

Note: 

8! = 8.7! = 8.7.6! (and so on) 
also, 7.6! = 7! 

n! = n(n - 1)! = n(n - 1)(n - 2)! (and so on) 
also, (n + 1)n! = (n + 1)! 

so in the case of yours: 

=> E (n = 0 -> 4) 1/n! = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! 

=> E (n = 0 -> 4) 1/n! = 1 + 1 + 1/2 + 1/6 + 1/24 

=> E (n = 0 -> 4) 1/n! = (24 + 24 + 12 + 4 + 1) / 24 

=> E (n = 0 -> 4) 1/n! = 65/24 

hope this helps :)