#1666 · Algebra · Inequalities

Growth of Exponential Expressions

If $α$ and $β$ are positive quantities, and $a > β$ , show that $(1 + \frac{1}{α})^{α} > (1 + \frac{1}{β})^{β}$ . Hence show that if $n > 1$ the value of $(1 + \frac{1}{n})^{n}$ lies between $2$ and $2.718 \dots$

~20 minProof · checked by AIInequalitiesCalculusLimits

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIX. Inequalities, Examples XIX.b, No. 4

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