Proving Inequality: [n/(n+1)]n ≤ [(n+1)/(n+2)]n+1

[Jamin2112]

Homework Statement

In the middle of a certain problem I am trying to prove that [n/(n+1)]ⁿ ≤ [(n+1)/(n+2)]ⁿ⁺¹.

Homework Equations

Could use induction (show it's true for n=1, and that if it's true for n, it's also true for n+1) or just start with some simple and obviously true inequality and then mess with it until I get the desired inequality.

The Attempt at a Solution

Haven't gotten anywhere.

 

[╔(σ_σ)╝]

I quickly looked from some other "simple" inequality but I didn't find one right away.

What I did find is that you can get rid of some terms then get a polynomial and use induction on it.

The induction on the polynomial is easier than the original inequality.

 

[Jamin2112]

I quickly looked from some other "simple" inequality but I didn't find one right away.

What I did find is that you can get rid of some terms then get a polynomial and use induction on it.

The induction on the polynomial is easier than the original inequality.

How so? Use the binomial expansion?

 

[╔(σ_σ)╝]

How so? Use the binomial expansion?

Multiplication and approximation.

BTw I still have exponents.

Basically, I arrived at a point where all I had to do was show that

(n+2)^n < (n^2 +2n +1)^(n-1)