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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.
However, Abel's theorem states that if the series is convergent for some value z such that | z – c | = r, then the sum of the series for x = z is the limit of the sum of the series for x = c + t (z – c) where t is a real variable less than 1 that tends to 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.
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
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F(x, F(y,z)) = F(F(x,y), z) (associativity). The simplest example is the additive formal group law F(x, y) = x + y. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin.
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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).