An $\mathrm{e}$ Bound, Refined

★ 7/10calculus Monotonicity Estimation Logarithms inequality

Problem

Let $a \in \mathbb{R}$ and

f(x) = \left(1 + \frac{a}{x}\right)\left(1 + \frac{1}{x}\right)^x, \qquad x > 0.

1. When $a = 1$ , prove that $f(x) > \mathrm{e}$ . 2. If $f(x) > \mathrm{e}$ holds for all $x > 0$ , find the range of possible values of $a$ . 3. For integers $n \geqslant 2$ , prove that

\frac{3}{2} < \left(1 + \frac{1}{2n}\right)^n < \frac{5}{3}.

Answer

Solution

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

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