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Three Proofs of the Inequality $e < {\left( {1 + \frac{1}{n}} \right)^{n + 0.5}}$

Sanjay K. Khattri
The American Mathematical Monthly
Vol. 117, No. 3 (March 2010), pp. 273-277
DOI: 10.4169/000298910x480126
Stable URL: http://www.jstor.org/stable/10.4169/000298910x480126
Page Count: 5
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Three Proofs of the Inequality $e < {\left( {1 + \frac{1}{n}} \right)^{n + 0.5}}$
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Abstract

Abstract The inequality $e > {(1 + 1/n)^n}$ is well known. In this work, we give three proofs of the inequality $e > {(1 + 1/n)^{n + 0.5}}$. For deriving the inequality, we use the Taylor series expansion and the Hermite-Hadamard inequality. In the third proof, we define a strictly increasing function which is bounded from above by 0.5.

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