Relation between $x^{x+1}$ and $(x+1)^{x}$, $x \in \mathbb{Z}$

Question

So say that we have a pair $(x^{x+1},(x+1)^{x})$ for all $x \in \mathbb{Z}$.

Is there any correlation between the members of this pair? Or are they not related?

Answer 1

The ratio $\frac{(x+1)^x}{x^{x+1}}=\frac{1}{x}\frac{(x+1)^x}{x^{x}}=\frac{1}{x}(1+\frac{1}{x})^x$

As $x$ gets large, $(1+\frac{1}{x})^x \rightarrow e$, so the ratio gets close to $\frac{e}{x}$ which itself gets closer to $0$ as $x$ increases.

Answer 2

Another way of interpreting Keith Blackman's answer above is that:

(x+1)^x>x^(x+1) ---(1)

Above equation (1) attains equality at x~2.2932

But 'x' above is not an integer.

There is a equation which has integer solution shown below.

x^y=y^x

Above has solution, (x,y)=(4,2)