A divergent series is a mathematical series that does not converge to a finite limit as the number of terms goes to infinity. In other words, the sum of the series either approaches infinity or oscillates without settling on a specific value. Mathematically, a series is said to be divergent if it does not satisfy the convergence condition:
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.
While divergent series do not converge to a finite value, they can still be useful in certain contexts. For example, they can sometimes be used to obtain asymptotic approximations, which provide insight into the behavior of a function or a sequence as it approaches a certain limit.