Bernoulli's Inequality

The bound (1+x)^n ≥ 1 + nx, its induction proof, and how it seeds AM-GM.

The idea

Theorem (Bernoulli's inequality).

For every real $x \ge -1$ and every integer $n \ge 0$, $(1+x)^{n} \;\ge\; 1 + nx.$ For $n \ge 2$ and $x \neq 0$ the inequality is strict; equality holds when $n = 0$, $n = 1$, or $x = 0$.

The right side consists of the first two terms of the binomial expansion of $(1+x)^{n}$, so the inequality says that the power never falls below its linear part. Geometrically, $y = 1 + nx$ is the tangent line to the curve $y = (1+x)^{n}$ at $x = 0$, and the curve stays on or above that line throughout $x \ge -1$. This is an instance of a general picture, drawn schematically in the figure: a curve $f$ that bends upward stays on or above the line touching it at any point $a$.

Ways to work on it

• Walkthrough. The bound, a concrete check, and the induction step.
• Practice. Evaluate the guaranteed lower bound for a given case.
• Hardest. Use the inequality to bound a root with no calculator.

Not sure where to start? Take the ten-question placement test.