# How to prove that $\left(1+\frac 1 n\right)^n < \left(1+\frac 1 {n+1}\right)^{n+1}$? [duplicate]

How would one prove the following:$$\left(1+\frac 1 n\right)^n < \left(1+\frac 1 {n+1}\right)^{n+1}$$

This is taken from the book challenge and thrill of precollege mathematics.

• Is $n\in\Bbb N$? Also, what have you tried? – Dave Jun 29 '17 at 11:47

One sledgehammer approach is to use calculus to verify that $x \ln(1+1/x)$ is monotonically increasing with $x>0.$
This is equivalent to showing that $(1+\frac{1}{n})^n$ is monotonically increasing. Here are many proofs:
I have to show $(1+\frac1n)^n$ is monotonically increasing sequence