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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.
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 ...
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 ...
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).
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [8] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
This differential equation has regular singular points at x = ±1 so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for | x | < 1 in general. When n is an integer, the solution P n (x) that is regular at x = 1 is also regular at x = −1, and the series for this solution ...
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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.