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The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
A series may also be represented with capital-sigma notation: [8] ... In general, a geometric series with initial term and common ratio , =, converges ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In capital-sigma notation this is expressed = or = + with a n > 0 for all n. Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit.
This series can be written by using sigma notation, as in the right side formula. [1] ... The Maclaurin series of 1 / 1 − x is the geometric series
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
(Capital Greek letter delta—not to be confused with , which may denote a geometric triangle or, alternatively, the symmetric difference of two sets.) 1. Another notation for the Laplacian (see above). 2. Operator of finite difference. or