Roots of $\tan x = x$

★ 8/10trigonometry calculus Monotonicity Trigonometric Identities Estimation inequality

Problem

List all positive roots of the equation $\tan x = x$ in increasing order, and let $r_n$ denote the $n$ -th one. Prove that for every positive integer $n$ ,

0 < r_{n+1} - r_n - \pi < \frac{1}{(n^2 + n)\pi}.

Answer

Solution

Difficulty	8/10
Topics	trigonometry, calculus, Monotonicity, Trigonometric Identities, Estimation, inequality

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