1. The problem statement, all variables and given/known data Prove: n! >or= (n^n)e^(1-n) Edit: For some positive integers n. (I'm very sure its all) 3. The attempt at a solution Proof by Induction: Base: n=1 1 >or= (1^1)e^0 = 1 Induction Step (n->n+1) (n+1)! = (n+1)n! >= (n+1)(n^n)e^(1-n) = (n+1)(n^n)ee^(-n) Now I want to show (n+1)(n^n)e is greater or equal to (n+1)^(n+1) Taking natural log of both sides we get: ln(n+1) + nln(n) + 1 >=? (n+1)ln(n+1) nln(n) + 1 >=? nln(n+1) ne^(1/n) >=? n+1 e >=? (1+1/n)^n Here at the very end of my proof is where I get stuck. I know that e can be defined as the limit as n approachs infinity of (1+1/n)^n but I'm unsure how to show that its greater then the sequence at all specific values of n. Any help would be appreciated.