What does it mean for a series to converge?

A series is said to converge when the sum of its terms approaches a specific, finite value as more and more terms are added. This occurs when, as you sum additional terms of the series, the total gets closer and closer to this limit, rather than growing indefinitely or fluctuating. Convergence implies stability in the total sum, meaning that no matter how many terms are included, you can find a clear, defined value it nears.

For example, consider a simple geometric series where each term decreases progressively, resulting in a sum that approaches a particular number instead of increasing indefinitely. This behavior contrasts sharply with divergence, where the series may grow without bound or fluctuate endlessly without settling on a specific value. Therefore, the essence of convergence captures the idea that a series will settle at a specific finite limit as you incorporate more terms, highlighting the predictable, stable nature of a convergent series.