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## Homework Statement

Prove: n! >or= (n^n)e^(1-n)

Edit: For some positive integers n. (I'm very sure its all)

## 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.

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