series

series in mathematics, indicated sum of a sequence of terms. A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g., the infinite series 1/2 + 1/4 + 1/8 + 1/16 + … . The dots mean that the remaining terms are formed according to the rule made evident by the first few terms, in this case doubling the denominator of the preceding term to form that of the next term; the n th term of this series is ( 1/2 ) ⁿ . Some infinite series converge to a certain value called its limit; i.e., as one adds together progressively more terms, these sums (called the partial sums of the series) form a sequence of values that progressively approach the limit. For example, the series given above converges to the value 1 because the partial sums form the sequence 1/2 , 3/4 , 7/8 , 15/16 , … . Many series, however, do not converge, i.e., have no value that their partial sums approach. Such a series is 1/2 + 1/3 + 1/4 + … , for even though the terms become very small, enough of them added together will give a value greater than any number that can be named. A series that does not converge is said to diverge; various tests exist for determining whether or not a given series converges and for determining its limit if it does converge. See also progression .

series Mathematical expression obtained by adding the terms of a sequence. Thus, the series 1 + 4 + 9 + 16 +… is formed from the sequence 1, 4, 9, 16…. Series may be finite or infinite, and infinite series may converge. An infinite series that fails to converge is said to diverge. A series formed from increasing powers of a variable is a power series; convergent power series are used for representing many functions.

series The major subdivision of a system, and the chronostratigraphic equivalent of an epoch. It denotes the layers of strata or the body of rock formed during one epoch. A series may itself be divided into stages. When used formally the initial letter of the term is capitalized, e.g. Lower Cretaceous Series.