How to Integrate Riemann’s Function — A Rigorous Step-by-Step Derivation

In this video, we study the integration of Riemann’s function defined by a highly oscillatory trigonometric series. Our goal is to compute the definite integral of this function over an arbitrary interval [x1, x2].We begin by justifying the interchange of summation and integration using the Weierstrass M-test and uniform convergence. Then we carefully perform the term-by-term integration and derive the final result. Timecode: 00:00 — Intro 00:24 — Problem & Solution: Defining and Integrating R(x) 04:58 — Geometric interpretation (Riemann’s function) Telegram: https://t.me/MathInfinitum TikTok:   / math.infinitum      • Theorem: Every Convergent Sequence is Boun...      • Solving Differential Equation xy\,dx + (x+...      • Does it Converge or Diverge? Integral of 1...      • Integral of 1/x^2: Power Rule with Negativ...      • Complex Analysis: Finding the Laurent Seri...      • Metric Spaces: Definition, Axioms, and Exa...      • Residue Calculation for Higher-Order and S...      • Evaluating Definite Complex Integral from ...      • Linear Differential Equation Solution: (xy...      • Method of Characteristics: Solving PDE (x(...      • Fourier Series Expansion of f(x) = x | Cal...      • Laplace Transform of f(t) = (cos(3t) - 1) ...   #riemann #function #integration #uniform #convergence #weierstrass #test #real #analysis #fourier_series #advanced #mathematical #series #basel #problem #lebesgue #math #lecture #mathematics #maths