Originally Posted by
terr13 Well, factorials grow at a larger rate than exponentials for large values of n. In fact, for n>=4, n, the sum of the factorials will always be larger than their exponents, so you can just do the other cases by hand. To prove that it is always larger, you can do induction, starting with n =4
4^2 =- 16
\sum_{1 4} n! = 24 + 6 + 2 +1 = 33
and show for 5, assume true for n, and show true for n +1.
Then you can look at 1, 2, 3.
1! = 1^2 = 1
2! = 2+1=3 != 2^2 = 4
3! = 6 + 2 + 1=9 = 3^2 = 9
So only 1 and 3.