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 nth 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.

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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