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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 .
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.
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
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, such as the computation of expected values in probability theory , especially in Bernoulli processes .
Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to a limit, in which case that limit is the value of the infinite series.