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Gregory Series — Definition, Formula & Examples

The Gregory Series (also called the Leibniz formula for π) is the infinite series 1−13+15−17+⋯=π41 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. It arises from evaluating the Maclaurin series for arctan⁡(x)\arctan(x) at x=1x = 1.

The Gregory Series is obtained by substituting x=1x = 1 into the power series representation arctan⁡(x)=∑n=0∞(−1)n2n+1x2n+1\arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}, valid for ∣x∣≤1|x| \le 1, yielding π4=∑n=0∞(−1)n2n+1\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}. The series converges conditionally but not absolutely.

Key Formula

π4=∑n=0∞(−1)n2n+1=1−13+15−17+⋯\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots
Where:
  • nn = Non-negative integer index of summation, starting at 0

How It Works

Start with the geometric series 11+t2=∑n=0∞(−1)nt2n\frac{1}{1+t^2} = \sum_{n=0}^{\infty} (-1)^n t^{2n} for ∣t∣<1|t| < 1. Integrate both sides from 00 to xx to obtain arctan⁡(x)=∑n=0∞(−1)n2n+1x2n+1\arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}. Abel's theorem justifies extending this to the endpoint x=1x = 1, giving π4=1−13+15−17+⋯\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots. Because the terms decrease in absolute value and tend to zero, the Alternating Series Test confirms convergence. However, the series converges very slowly — hundreds of terms are needed for even a few correct decimal places of π\pi.

Worked Example

Problem: Approximate π/4 using the first four terms of the Gregory Series and bound the error.
Step 1: Write out the first four partial sums (n = 0 through n = 3).
S3=1−13+15−17S_3 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}
Step 2: Compute the sum numerically.
S3=1−0.3333+0.2−0.1429≈0.7238S_3 = 1 - 0.3333 + 0.2 - 0.1429 \approx 0.7238
Step 3: By the Alternating Series Remainder theorem, the error is bounded by the absolute value of the first omitted term (n = 4).
∣error∣≤12(4)+1=19≈0.1111|\text{error}| \le \frac{1}{2(4)+1} = \frac{1}{9} \approx 0.1111
Answer: The four-term approximation gives π/4 ≈ 0.7238, compared to the true value π/4 ≈ 0.7854. The error is at most 1/9 ≈ 0.111.

Why It Matters

The Gregory Series is one of the earliest known representations of π as an infinite series and serves as a standard example when studying power series integration and conditional convergence in Calculus II. It also illustrates why mathematicians sought faster-converging series for computing π, motivating techniques like Machin's formula.

Common Mistakes

Mistake: Assuming the Gregory Series converges absolutely because it converges.
Correction: The series of absolute values is ∑12n+1\sum \frac{1}{2n+1}, which diverges (it behaves like the harmonic series). The Gregory Series converges only conditionally.