For any positive integer n, define fₙ : (0, ∞) → ℝ as fₙ(x) = Σⱼ₌₁ⁿ tan⁻¹(1/(1 + (x + j)(x + j − 1))) for all x ∈ (0, ∞). (Here, the inverse trigonometric function tan⁻¹x assumes values in (−π/2, π/2).) Then, which of the following statement(s) is (are) TRUE?