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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted. The n th partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if and only if the sequence of its partial sums tends to a limit ...
The geometric series is an infinite series derived from a special type of sequence called a geometric progression, which is defined by just two parameters: the initial term and the common ratio . Finite geometric series have a third parameter, the final term's power.
t. e. In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence ...
Radius of convergence. In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal ...
Formula for primes. In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.
The extremely slow convergence of the arctangent series for | | makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed 1 4 π {\displaystyle {\tfrac {1}{4}}\pi } as a sum of arctangents of smaller values, eventually resulting in a variety of ...
Escape time algorithm. The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x, y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.
The exponential of X, denoted by eX or exp (X), is the n×n matrix given by the power series. where is defined to be the identity matrix with the same dimensions as . 1 The series always converges, so the exponential of X is well-defined. Equivalently, where I is the n×n identity matrix. When X is an n×n diagonal matrix then exp (X) will be ...