### Topic Page: series

**series**from

*Philip's Encyclopedia*

**series**

*The Columbia Encyclopedia*

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.

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##### Full text Article series

The sum of the terms in a sequence , written as a 1 + a 2 + a 3 + a r + . The partial sum to the n th term is denoted by S n ...

##### Full text Article partial sum (of an infinite series)

Any sum of a finite number of consecutive terms in an infinite series , starting with the first. For the series a 1 + a 2 + … + an + … ...

##### Full text Article sum

1. The result of an addition . 2. (of sets) See union . 3. (of an infinite series) The limit of the sequence of ...