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A power series = is convergent for some values of the variable x, which will always include x = c since () = and the sum of the series is thus for x = c. The series may diverge for other values of x , possibly all of them.
When the variable is a positive integer, the number () is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size .
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell . Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)} , called the Bell series ...
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.
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
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If X is a discrete random variable taking values x in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1] G ( z ) = E ( z X ) = ∑ x = 0 ∞ p ( x ) z x , {\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x},} where p {\displaystyle p} is the probability mass ...