Convergence of a series refers to the behavior of the terms in the series as the number of terms goes to infinity. A series is said to converge if the terms approach a finite limit. On the other hand, divergence of a series occurs when the terms do not approach a finite limit as the number of terms increases.
There are several methods for testing convergence or divergence of a series, including the divergence test, comparison test, ratio test, and integral test. These methods involve comparing the given series to another known series or using mathematical operations to determine the behavior of the terms.
The convergence or divergence of a series tells us whether the terms in the series become infinitely large or approach a finite limit. This information is important in determining the behavior and characteristics of a series, as well as its applications in fields such as mathematics, physics, and engineering.
A series is said to have absolute convergence if the series of absolute values of its terms converges. On the other hand, conditional convergence occurs when the series converges, but the series of absolute values of its terms diverges. In other words, for conditional convergence, the series must be rearranged in a specific order to converge.
No, a series cannot be both convergent and divergent. A series can either converge or diverge, depending on the behavior of its terms. However, it is possible for a series to have both convergent and divergent subsequences, which can occur in alternating series or series with oscillating terms.