A finite series is written as a1+a2+…+an, where a1, a2,…, an are n numbers called the terms in the series, and n, some positive integer, is the length of the series. The sum of the series is simply the sum of the n terms. For certain finite series, such as arithmetic series and geometric series, the sum of the series is given by a known formula. The following can also be established:
An infinite series is written as a1+a2+a3+…, with terms a1, a2, a3,…, one corresponding to each positive integer. Let sn be the sum of the first n terms of such a series. If the sequence s1, s2, s3,…has a limit s, then the value s is called the sum (or sum to infinity) of the infinite series. Otherwise, the infinite series has no sum. See also arithmetic series, geometric series, binomial series, Taylor series and Maclaurin series.