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In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values. One way to view a formal power series is as an infinite ordered sequence of numbers. In this case, the powers of the variable are used only to indicate the order of the coefficients. The coefficient of is just the fifth term in the series, while the coefficient of is the zeroth term. In combinatorics, formal power series provide representations of numerical sequences and of multisets, and for instance allow concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring. ==Introduction== 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). For example, consider the series : If we studied this as a power series, its properties would include, for example, that its radius of convergence is 1. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients (−3, 5, −7, 9, −11, ... ). In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the factorials (1, 2, 6, 24, 120, 720, 5040, … ) as coefficients, even though the corresponding power series diverges for any nonzero value of ''X''. Arithmetic on formal power series is carried out by simply pretending that the series are polynomials. For example, if : then we add ''A'' and ''B'' term by term: : We can multiply formal power series, again just by treating them as polynomials (see in particular Cauchy product): : Notice that each coefficient in the product ''AB'' only depends on a ''finite'' number of coefficients of ''A'' and ''B''. For example, the ''X''5 term is given by : For this reason, one may multiply formal power series without worrying about the usual questions of absolute, conditional and uniform convergence which arise in dealing with power series in the setting of analysis. Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series ''A'' is a formal power series ''C'' such that ''AC'' = 1, provided that such a formal power series exists. It turns out that if ''A'' has a multiplicative inverse, it is unique, and we denote it by ''A''−1. Now we can define division of formal power series by defining ''B''/''A'' to be the product ''BA''−1, provided that the inverse of ''A'' exists. For example, one can use the definition of multiplication above to verify the familiar formula : An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator for a formal power series in one variable extracts the coefficient of ''Xn'', and is written e.g. ()(''A''), so that ()(''A'') = 5 and ()(''A'') = −11. Other examples include : Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Formal power series」の詳細全文を読む スポンサード リンク
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