A hypergeometric series is a type of power series that generalizes the concept of geometric series. It is defined by a specific ratio of factorials and can be expressed in the form {}_pF_q, where p and q are non-negative integers. These series often arise in combinatorics, number theory, and mathematical physics, providing solutions to various problems. The hypergeometric series converges under certain conditions and can represent many functions, including polynomials and exponential functions. It plays a crucial role in the study of special functions, such as the Legendre polynomials and Bessel functions, which have applications in physics and engineering.