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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).
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for ...
The Hilbert–Poincaré series is a formal power series used to study graded algebras. Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition , multiplication , derivative , antiderivative for power series "formally", treating the symbol ...
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 ...
The power series Q is called the characteristic power series of the genus . A theorem of René Thom , which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4 k for positive integers k , implies that this gives a bijection between formal power series Q with rational coefficients and ...
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.