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John Wallis, English mathematician who is given partial credit for the development of infinitesimal calculus and pi. Viète's formula, a different infinite product formula for. π {\displaystyle \pi } . Leibniz formula for π, an infinite sum that can be converted into an infinite Euler product for π. Wallis sieve.
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)
Digamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: [ 1][ 2][ 3] It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , [ 4] and it asymptotically behaves as [ 5] for complex numbers with large modulus ( ) in the sector with ...
Series expansion. An animation showing the cosine function being approximated by successive truncations of its Maclaurin series. In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating a function that cannot be expressed by just elementary ...
Cauchy's convergence test. The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821. [1]
Generating function. In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an ...
The Cauchy product can be defined for series in the spaces ( Euclidean spaces) where multiplication is the inner product. In this case, we have the result that if two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.
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 coefficient and the common ratio .