The Arithmetic-Geometric Mean Inequality and the Constant $e$.

by Hansheng Yang + Heng Yang

This article originally appeared in:
Mathematics Magazine
October, 2001

Subject classification(s): Calculus | Single Variable Calculus
Applicable Course(s): 3.2 Mainstream Calculus II | 3.5 Non-mainstream Calc II

In this note the authors present an elementary proof that the inequalities $(1+1/n)^n < e \leq (1+1/(m-1))^m$ hold for $n>0$ and $m>1$, using only the arithmetic-geometric mean inequality.

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Capsule Course Topic(s):
One-Variable Calculus | Infinite Limits: Function Values and Integrals