Is n! Less Than (n+1)^n Simplifiable?

[brookey86]

Homework Statement

n! < (n+1)ⁿ

I am not looking for a proof, just a way to simplify this equation mathematically.

Homework Equations

The Attempt at a Solution

As far as I can see, I cannot simplify this any further. Is there something I can divide out of both sides, for example?

 

[zcd]

$$n! = \prod_i=1^n i$$
$$(n+1)^n =\prod_i=1^n (n+1)$$

What can you say about the relation between i and n+1?

 

[zcd]

$$n! = \prod_i=1^n i$$
$$(n+1)^n =\prod_i=1^n (n+1)$$

What can you say about the relation between i and n+1?

EDIT: bad latex
$$n! = \prod_{i=1}^n i$$
$$(n+1)^n =\prod_{i=1}^n (n+1)$$

 

[brookey86]

I can see that i will always be less than n+1. But is there a way to compare the two without using the product summation symbol?

 

[zcd]

You could confirm what you wrote:
$$(\frac{1}{n+1})( \frac{2}{n+1} ) ...( \frac{n-1}{n+1} )( \frac{n}{n+1} )< 1 \Rightarrow n! < (n+1)^n$$

 

[HallsofIvy]

Since that formula depends upon the positive integer n, you might consider proof by induction.