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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.
This is a special case of the generalization of a geometric series of real or complex numbers to a geometric series of operators. The generalized initial term of the series is the identity operator T 0 = I {\displaystyle T^{0}=I} and the generalized common ratio of the series is the operator T . {\displaystyle T.}
The formula above is a geometric series—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the sum formula for a geometric series. Archimedes evaluates the sum using an entirely geometric method, [8] illustrated in the adjacent picture. This picture shows a unit square which has ...
14th century - Madhava discovers the power series expansion for , , and / [6] [7] This theory is now well known in the Western world as the Taylor series or infinite series. [ 8 ] 14th century - Parameshvara discovers a third order Taylor interpolation for sin ( x + ϵ ) {\displaystyle \sin(x+\epsilon )} ,
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method assigns = to / for all in a subset of the complex plane, given certain restrictions on , then the method also gives the analytic continuation of any other function () = = on ...
In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series x n is called hypergeometric if the ratio of successive terms x n+1 /x n is a rational function of n.
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