A Log Telescoping Bound

★ 8/10calculus Monotonicity Estimation Telescoping inequality

Problem

Let $f(x) = x\ln x - ax^2 + a$ with $a > 0$ .

1. If $f(x) < 0$ for all $x > 1$ , find the range of possible values of $a$ . 2. For integers $n \geqslant 2$ , prove that

\frac{\ln 2}{3\cdot 5} + \frac{\ln 3}{5\cdot 7} + \cdots + \frac{\ln n}{(2n-1)(2n+1)} < \frac{1}{4}.

Answer

Solution

Difficulty	8/10
Topics	calculus, Monotonicity, Estimation, Telescoping, inequality

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