Gregory Series — Definition, Formula & Examples

The Gregory Series (also called the Leibniz formula for π) is the infinite series

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}

. It arises from evaluating the Maclaurin series for

\arctan(x)

x = 1

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

Key Formula

\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots

Where:

$n$ = Non-negative integer index of summation, starting at 0

How It Works

Start with the geometric series

\frac{1}{1+t^2} = \sum_{n=0}^{\infty} (-1)^n t^{2n}

for

|t| < 1

. Integrate both sides from

0

x

to obtain

\arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} x^{2n+1}

. Abel's theorem justifies extending this to the endpoint

x = 1

, giving

\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

\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).

S_3 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7}

Step 2: Compute the sum numerically.

S_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).

|\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

\sum \frac{1}{2n+1}

, which diverges (it behaves like the harmonic series). The Gregory Series converges only conditionally.

Related Terms

Alternating Series — The Gregory Series is an alternating series
Alternating Series Test — Proves convergence of the Gregory Series
Alternating Series Remainder — Bounds the error of partial sums
Conditional Convergence — Gregory Series converges conditionally, not absolutely
Integral of a Power Series — Series derived by integrating a geometric series
Absolute Convergence — Gregory Series fails absolute convergence