Rearrangement Question

Can anybody help out with showing how we can rearrange the following equation.

How do you rearrange (1+1/n)^n < (1/(n+1))^(n+1) to get (n(n+2)/(n+1)^2)^(n+1) > n/(n+1)

Do you mind taking a picture of the question/latex it, I can barely comprehend it.

What's in the bracket in the first inequality, RHS?

It says (1 + something and the bracket never closes and it doesn't seem to resemble the original thing you typed in your OP.

On the bracket on the RHS its meant to be (1+1/(n+1))^(n+1)

In the latex version, ignore the small open bracket, thats just a typo

Okay, so we have $(1+\frac{1}{n})^n \geq (1+\frac{1}{n+1})^{n+1}$

This is the same thing as $(\frac{n}{n+1})^n \geq ((\frac{n+2}{n+1})^{n+1}$

Inverting both sides and changing the inequality sign

$(\frac{n}{n+1})^n \leq (\frac{n+1}{n+2})^{n+1}$

Multiplying both sides by $(\frac{n+1}{n})^n$ yields

$1 \leq (\frac{(n+1)^2}{n(n+2)})^n \cdot \frac{n+1}{n+2}$

Can you take it from there? It'll require another reciprocation and some manipulation.

Sorry for the messy latex, in a rush

Shouldn't it be $1 + \frac{1}{n} = \frac{n+1}{n}$? After which the result follows immediately by multiplying by a suitable value (assuming that said value is positive).

EDIT: Never mind, it seems you have put the fraction the wrong way up in your second line but corrected it later.