A divergent series is a series in which the terms do not approach a finite limit, meaning that the sum of the series does not converge to a specific value. In contrast, a convergent series is a series in which the terms approach a finite limit, resulting in a sum that converges to a specific value.
There are several methods for determining if a series is divergent or convergent, including the comparison test, the integral test, and the ratio test. These methods involve comparing the given series to a known series or using properties of integrals to determine convergence or divergence.
Divergent and convergent series have many applications in fields such as physics, engineering, and finance. For example, in physics, series are used to approximate functions and calculate values such as energy and velocity. In finance, series are used in financial modeling and risk analysis.
No, a series can only be either divergent or convergent. If a series is divergent, it cannot also be convergent, and vice versa.
Divergent series are often associated with the concept of infinity, as the terms of the series do not approach a finite limit. In contrast, convergent series are associated with finite values and do not extend to infinity. However, it is important to note that not all infinite series are divergent, as some can converge to a finite value.