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

Question

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 — Dave, Jun 29 '17 at 11:47

score 3 · Answer 1 · answered Jun 29 '17 at 11:47

3

One sledgehammer approach is to use calculus to verify that $x \ln(1+1/x)$ is monotonically increasing with $x>0.$

answered Jun 29 '17 at 11:47

Reiner Martin

Perhaps this problem is intended as a pre-log exercise? – Bernard Jun 29 '17 at 11:57
Probably, that's why I called it a sledgehammer approach. There are plenty of nice elementary solutions available at the link given by dromastyx. – Reiner Martin Jun 29 '17 at 12:04

score 0 · Answer 2 · answered Jun 29 '17 at 11:51

0

This is equivalent to showing that $(1+\frac{1}{n})^n$ is monotonically increasing. Here are many proofs:

answered Jun 29 '17 at 11:51

dromastyx

2 Answers2