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Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is , , , , , … where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series.
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 nth element of an arithmetico-geometric sequence is the product of the nth element of an arithmetic sequence and the nth element of a geometric sequence. [1] An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications ...
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
Geometric progression, a sequence of numbers such that the quotient of any two successive members of the sequence is a constant; Harmonic progression (mathematics), a sequence of numbers such that their reciprocals form an arithmetic progression; In music: Chord progression, series of chords played in order
Using comparisons to geometric series specifically, [20] [21] those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms.
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.
An integer sequence is a sequence whose terms are integers. A polynomial sequence is a sequence whose terms are polynomials. A positive integer sequence is sometimes called multiplicative, if a nm = a n a m for all pairs n, m such that n and m are coprime. [8] In other instances, sequences are often called multiplicative, if a n = na 1 for all n.