Series

In mathematics, a series can be generally described as the sum of an infinite sequence of values. Series are largely used in calculus, as well as in other areas of mathematics, physics, computer science, statistics, and finance.


Series notation


The values of a sequence, the sum of which form a series, are referred to as terms or elements. Terms can be numbers, functions, or essentially anything that can be added. "Series" and "infinite series" are often used interchangeably. The "infinite" in infinite series is meant to emphasize that the series contains an infinite number of terms. Since finite series are not usually considered in the context if calculus, any use of the word "series" on this site will mean infinite series unless otherwise specified. Series are commonly represented in two ways: as a sum of variables followed by an ellipsis (...) or with the use of the summation sign, as shown below:

  1. , where the subscript denotes which term in the series is being represented
  2. . This reads as "the sum from i=1 to ∞ of ai."

Series can be convergent or divergent. When a series is divergent, the sum of the series cannot be computed. When it is convergent, the series is said to be summable (specifically the sequence is summable), and a value can be assigned to the series using the following limit: