A convergent series is one in which the sum of its terms approaches a finite value as the number of terms increases. In contrast, a divergent series is one in which the sum of its terms does not approach a finite value as the number of terms increases.
One way to determine if a series is convergent or divergent is to use the convergence tests, such as the ratio test, integral test, or comparison test. These tests involve checking the behavior of the terms of the series to determine if they approach a finite value or not.
Knowing if a series is convergent or divergent is important in many areas of mathematics and science. It allows for the calculation of limits, the approximation of values, and the evaluation of integrals. In addition, the convergence or divergence of a series can provide insights into the behavior and properties of the underlying function.
No, a series cannot be both convergent and divergent. A series can either have a finite sum (convergent) or have an infinite sum (divergent). It cannot have both at the same time.
Convergent and divergent series have various applications in fields such as physics, engineering, and economics. For example, in physics, they are used to calculate the total energy of a system. In engineering, they are used to approximate complex systems and in economics, they are used to model the growth of populations or investments.