What is Catalan's Constant?

Alternating sum 1 − 1/9 + 1/25 − … converging to G

G = 1 − 1/9 + 1/25 − 1/49 + … = Σ (−1)ⁿ/(2n+1)². The alternating series converges slowly. Whether G is irrational remains unknown.

Three equivalent forms of Catalan's constant

G = Σₙ₌₀^∞ (−1)ⁿ/(2n+1)² ≈ 0.91597…

G = ∫₀¹ arctan(t)/t dt = ∫₀^(π/2) ln(1/sin t)/2 · dt

All three expressions are equal. G appears in combinatorics, physics, and analysis.

Related topics

Basel Problem Apery Wallis Product

Apéry's Constant → Erdős-Borwein Constant → Riemann Zeta Function → Euler-Mascheroni Constant →

Key facts about Catalan's Constant

Catalan's constant G = 1 - 1/9 + 1/25 - 1/49 + ... = 0.91596559... Whether it is irrational is one of the major open problems in mathematics. It appears in combinatorics, in evaluating certain integrals, and as the value of the Dirichlet beta function at 2. Studied by Eugène Catalan in 1865. Computed to over 600 billion decimal places.

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Question

What is a clean integral representation of Catalan's constant?

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