A series is a sum of an infinite sequence of numbers, typically denoted by the sigma notation ∑. Each term in the sequence is added together to determine the value of the series.
A convergent series is one whose terms approach a finite value as the number of terms increases. In other words, the sum of the series is a finite number. On the other hand, a divergent series is one whose terms do not approach a finite value and the sum of the series is either infinity or negative infinity.
To determine the convergence or divergence of a series, we can use various tests such as the ratio test, the root test, or the integral test. These tests compare the terms of the series to well-known sequences that either converge or diverge, ultimately determining the convergence or divergence of the series.
Series and convergence/divergence have various applications in mathematics, especially in calculus and analysis. They are used to approximate functions, evaluate integrals, and solve differential equations. Additionally, they are important in understanding the behavior of infinite sequences and the concept of infinity.
No, a series cannot be both convergent and divergent. A series can only have one of these properties. However, it is possible for a series to be conditionally convergent, meaning that it is convergent but not absolutely convergent. In other words, the series converges but the absolute values of the terms do not converge.