what is a geometric series

A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. It is written as a+ar+ar2+ar3+…a+ar+ar^2+ar^3+\ldots a+ar+ar2+ar3+…, where (a) is the coefficient of each term and (r) is the common ratio between adjacent terms. The series converges if and only if the absolute value of the common ratio is less than one (|r| < 1) and its value can be computed using the formula a1−r\frac{a}{1-r}1−ra​. Geometric series are used in various mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential