A sum of an infinite series is the total value obtained by adding an infinite number of terms in a specific order. It is represented by the symbol ∑ (sigma) and is used to represent the sum of all the terms in the series.
The sum of an infinite series can be calculated by finding the limit of the partial sums as the number of terms approaches infinity. This is known as the convergence of the series. If the limit exists, the series is said to be convergent and the limit is the value of the sum. If the limit does not exist, the series is said to be divergent and the sum is undefined.
A convergent series is one in which the limit of the partial sums exists and is a finite number. This means that the sum of the series can be calculated and has a definite value. On the other hand, a divergent series is one in which the limit of the partial sums does not exist, and therefore, the sum of the series is undefined.
Some common methods for determining the convergence of a series include the comparison test, ratio test, root test, and integral test. These methods involve comparing the given series to a known series with known convergence properties to determine the convergence or divergence of the given series.
Yes, there are several real-life applications of calculating sums of infinite series. For example, in finance, the present value of a continuous stream of cash flows can be calculated using infinite series. In physics, infinite series are used to calculate the position and velocity of an object moving at a constant acceleration. In computer science, infinite series are used in algorithms for data compression and signal processing.