A convergent series is a mathematical series whose terms approach a finite limit as the number of terms goes to infinity. In other words, the sum of the series converges to a specific value. Mathematically, a series is said to be convergent if:
lim (n → ∞) S_n = S
where S_n is the partial sum of the first n terms of the series, and S is the limit of the series.
Convergent series play an important role in calculus, particularly in the context of power series and their applications, such as the Taylor series and the Maclaurin series. These series are used to approximate functions and solve differential equations, among other applications.