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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 .
A Neumann series is a mathematical series that sums k ... series of real or complex numbers to a geometric series of ... approximation to the ...
Comparison of the convergence of two Madhava series (the one with √ 12 in dark blue) and several historical infinite series for π. S n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)
where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and , are the arithmetic and geometric iterations of (,), the arithmetic-geometric mean of a and b with the initial values = and =.
Laurent series – Power series with negative powers; Padé approximant – 'Best' approximation of a function by a rational function of given order; Newton series – Discrete analog of a derivative; Approximation theory – Theory of getting acceptably close inexact mathematical calculations
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that = + + = = +,. an alternating series.. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), [1] and was later independently rediscovered by James Gregory in ...
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.
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