The Gregory series is a mathematical series used to approximate the value of π (pi). It is derived from the Taylor series expansion of the arctangent function. The series is expressed as: \[ \frac\pi4 = 1 - \frac13 + \frac15 - \frac17 + \frac19 - \ldots \] This alternating series converges to π/4, meaning that as more terms are added, the approximation becomes more accurate. The series is named after the mathematician James Gregory, who first introduced it in the 17th century. It is significant in the study of calculus and numerical methods, providing a simple way to calculate π using basic arithmetic operations.