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The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential.
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, non-zero number called the common ratio.
Low-order polylogarithms Finite sums: , ( geometric series) Infinite sums, valid for (see polylogarithm ): The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form : Exponential function (cf. mean of Poisson distribution) (cf. second moment of Poisson distribution) where is the Touchard ...
Euclid expresses the result by stating that if a finite geometric series beginning at 1 with ratio 2 has a prime sum q, then this sum multiplied by the last term t in the series is perfect. Expressed in these terms, the sum q of the finite series is the Mersenne prime 2p − 1 and the last term t in the series is the power of two 2p−1.
The geometric distribution is the discrete probability distribution that describes when the first success in an infinite sequence of independent and identically distributed Bernoulli trials occurs. Its probability mass function depends on its parameterization and support.
A finite sequence can be viewed as an infinite sequence with only finitely many nonzero terms, or in other words as a function with finite support. For any complex-valued functions f, g on with finite support, one can take their convolution : Then is the same thing as the Cauchy product of and .
Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression.