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Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
The order of the power series f is defined to be the least value such that there is a α ≠ 0 with = | | = + + +, or if f ≡ 0. In particular, for a power series f(x) in a single variable x, the order of f is the smallest power of x with a nonzero coefficient.
Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is =, has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function f(z) of Example 1 is the derivative of g(z). Example 3: The power series
This implies that the series converges for any finite value of z and thus defines an entire function of z. An example is the power series for the exponential function. If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 and diverges for |z| > 1.
Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.
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 ...
then (S n f)(x 0) converges to ℓ. This implies that for any function f of any Hölder class α > 0, the Fourier series converges everywhere to f(x). It is also known that for any periodic function of bounded variation, the Fourier series converges. In general, the most common criteria for pointwise convergence of a periodic function f are as ...