Series (mathematics)

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series. In modern terminology, any ordered infinite sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} of terms, whether those terms are numbers, functions, matrices, or anything else that can be added, defines a series, which is the addition of the ⁠ a i {\displaystyle a_{i}} ⁠ one after the other. To emphasize that there are an infinite number of terms, series are often also called infinite series to contrast with finite series, a term sometimes used for finite sums. Series are represented by an expression like a 1 + a...