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?

In the positives, both functions are invertible, the relation is immediate. — , May 09 '20 at 10:42
We need the trichotomy $$(x+1)^{1/x+1} >=< x^{1/x}$$ Use https://math.stackexchange.com/questions/116112/find-the-maximum-of-fx-x1-x — lab bhattacharjee, May 09 '20 at 11:00
Exactly what do you mean by correlation? Otherwise this question is so broad as to be unsuited for exact analysis. — Allawonder, May 09 '20 at 14:42

score 2 · Answer 1 · answered May 09 '20 at 18:52

2

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.

answered May 09 '20 at 18:52

Keith Backman

score 1 · Answer 2 · answered May 10 '20 at 05:27

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)

2 Answers2