Product Versus Exponent

★ 7/10calculus Monotonicity Logarithms inequality

Problem

A sequence $\{x_n\}$ ( $n \in \mathbb{N}^*$ ) satisfies $x_n > 1$ . Let $\mathrm{e}$ be the base of the natural logarithm, and define

A = x_1 x_2 \cdots x_n, \qquad B = x_1 + x_2 + \cdots + x_n, \qquad C = \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}.

Prove that $A^2 < \mathrm{e}^{B - C}$ .

Answer

Solution

Difficulty	7/10
Topics	calculus, Monotonicity, Logarithms, inequality

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