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Catalan's Constant — Definition, Formula & Examples

Catalan's constant, denoted GG, is a number approximately equal to 0.91596550.9159655 that arises from summing the alternating series of reciprocal odd squares. It appears frequently in combinatorics, number theory, and evaluations of definite integrals.

Catalan's constant is defined as G=∑n=0∞(−1)n(2n+1)2G = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}, which equals 112−132+152−172+⋯\frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots. It is equivalently expressed as G=−∫01ln⁡t1+t2 dtG = -\int_0^1 \frac{\ln t}{1+t^2}\,dt. Whether GG is irrational remains an open problem.

Key Formula

G=∑n=0∞(−1)n(2n+1)2=1−19+125−149+⋯G = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} = 1 - \frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \cdots
Where:
  • GG = Catalan's constant, approximately 0.9159655941772190
  • nn = Non-negative integer index of summation

Worked Example

Problem: Approximate Catalan's constant using the first four terms of its defining series.
Write out terms: Expand the series for n=0,1,2,3n = 0, 1, 2, 3.
G≈112−132+152−172G \approx \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2}
Evaluate each fraction: Compute the decimal value of each term.
G≈1−0.11111+0.04000−0.02041G \approx 1 - 0.11111 + 0.04000 - 0.02041
Sum the terms: Add the four values together.
G≈0.90848G \approx 0.90848
Answer: Using four terms gives G≈0.9085G \approx 0.9085, compared to the true value G≈0.9160G \approx 0.9160. The series converges slowly, so many more terms are needed for high precision.

Why It Matters

Catalan's constant appears as the closed-form answer to many definite integrals and lattice-counting problems. For instance, it equals the mean value of the Clausen function and arises in computing the number of alternating permutations. In physics, it shows up in exact solutions to certain statistical mechanics models on square lattices.

Common Mistakes

Mistake: Using all positive integers in the denominators instead of only odd squares.
Correction: The denominators are (2n+1)2(2n+1)^2: that is 1,9,25,49,…1, 9, 25, 49, \ldots — only odd numbers squared. Summing 1/n21/n^2 with alternating signs gives a different value related to π2/12\pi^2/12.