The harmonic series is a mathematical series defined as the sum of the reciprocals of the natural numbers. It can be expressed as 1 + 1/2 + 1/3 + 1/4 + ... and continues indefinitely. Although the terms of the series get smaller, the total sum grows without bound, meaning it diverges to infinity. This series is significant in various fields, including number theory and analysis. It also appears in applications such as physics and computer science, particularly in algorithms and complexity analysis. Understanding the harmonic series helps illustrate concepts of convergence and divergence in mathematics.