Christoffer De Geer March 21, 2026

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.

Used in

∑Mathematics

✓

⚛Physics

✓

⚙Engineering

-

🧬Biology

-

💻Computer Sci

-

📊Statistics

-

📈Finance

-

🎨Art

-

🏛Architecture

-

♪Music

-

🔐Cryptography

-

🌌Astronomy

-

⚗Chemistry

-

🦉Philosophy

-

🗺Geography

-

🌿Ecology

-

Want to test your knowledge?

Question

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

tap · space

1 / 10

Ready to play?

π

Pi

Memorize pi, e, and 38 mathematical constants using the numpad path method

Works on any device.

Topic roundups

#memory All memory games Hold a sequence, grid, or list in working memory and reproduce it under pressure. #numbers All numbers games Arithmetic, ratios, units, and number sense in short timed rounds. #sequence All sequence games Reproduce or extend an ordered string of items in the right order.