# Monotonic Convergence to $e$ via the Arithmetic-Geometric Mean

by Józef Sáandor (Babes-Bolyai University Romania)

This article originally appeared in:
Mathematics Magazine
June, 2007

Subject classification(s): Calculus | Single Variable Calculus | Limits
Applicable Course(s): 3.1 Mainstream Calculus I

Hansheng Yang and Heng Yang used the arithmetic-geometric mean inequality to prove that the sequence $[1+1/n]^n$ is monotonic increasing converging to $e$ whereas $[1+1/n]^{n+1}$ is monotonic decreasing converging to $e$. The author provides a simpler proof using the same technique.

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Capsule Course Topic(s):
Sequences and Series | Approximations: pi, e, natural logarithms