3

Use the AM - GM inequality (no other method is acceptable), to prove that for all positive integers $n$:

$$\left(1 +\dfrac{1}{n}\right)^n \leq \left(1 + \dfrac{1}{n+1}\right)^{n+1}$$

I see that it is increasing... Don't know how to keep going.

mathematics2x2life
  • 13,322

1 Answers1

6

One way is to rewrite the inequality as:

$$\sqrt[n+1]{\left(1+\frac{1}{n}\right)^n\cdot 1}\le \frac{n\left(1+\frac{1}{n}\right)+1}{n+1},$$ and note that this is precisely AM-GM for $\left(1+\frac{1}{n}\right)$ taken $n$ times and $1.$

Vincenzo Tibullo
  • 10,955
leshik
  • 4,850
  • So, Could I start by finding the AM-GM for (1+1/n) taken n times. Could I show by induction that it also for n+1. Then find AM-GM for n+1 too? –  Nov 11 '13 at 23:10