Infinite sum that is considered independently from any notion of convergence
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In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).
A formal power series is a special kind of formal series, whose terms are of the form where is the th power of a variable ( is a non-negative integer), and is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the are used only as position-holders for the coefficients, so that the coefficient of is the sixth 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 recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring.
Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to p-adic integers, which can be defined as formal series of the powers of p.
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operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formalpowerseries is a special kind of formalseries, whose terms...
in mathematical analysis, powerseries also occur in combinatorics as generating functions (a kind of formalpowerseries) and in electronic engineering...
{\displaystyle F[[x]]} of formalpowerseries. Puiseux series Mittag-Leffler's theorem Formal Laurent series – Laurent series considered formally, with coefficients...
determining cause Formalpowerseries, a generalization of powerseries without requiring convergence, used in combinatorics Formal calculation, a calculation...
{\displaystyle x} . Another important example of a DVR is the ring of formalpowerseries R = k [ [ T ] ] {\displaystyle R=k[[T]]} in one variable T {\displaystyle...
In mathematics, a formal group law is (roughly speaking) a formalpowerseries behaving as if it were the product of a Lie group. They were introduced...
being a not necessarily commutative ring, and with formal skew powerseries in place of formalpowerseries. There is also a Weierstrass preparation theorem...
obtained by the formal computation. Formalpowerseries is a concept that adopts the form of powerseries from real analysis. The word "formal" indicates that...
sequence of numbers as the coefficients of a formalpowerseries. Unlike an ordinary series, the formalpowerseries is not required to converge: in fact, the...
algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formalpowerseries. If K is a field (such as the...
easiest (though somewhat heavy) construction starts with the ring of formalpowerseries R [ [ X 1 , X 2 , . . . ] ] {\displaystyle R[[X_{1},X_{2},...]]}...
polynomials in characteristic classes that arise as coefficients in formalpowerseries with good multiplicative properties. A genus φ {\displaystyle \varphi...
The International Conference on FormalPowerSeries and Algebraic Combinatorics (FPSAC) is an annual academic conference in the areas of algebraic and...
also called reversion of series. If the assertions about analyticity are omitted, the formula is also valid for formalpowerseries and can be generalized...
David O. Tall, which are lexicographically ordered fractions of formalpowerseries over the reals. Suppose X is a Tychonoff space and C(X) is the algebra...
algebra, the ring of restricted powerseries is the subring of a formalpowerseries ring that consists of powerseries whose coefficients approach zero...
In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formalpowerseries that mimics the form of the derivative...
same as for polynomials. Non-formalpowerseries also generalize polynomials, but the multiplication of two powerseries may not converge. A polynomial...
generalization of Puiseux series (themselves a generalization of formalpowerseries) and were first introduced by Hans Hahn in 1907 (and then further...
the usual powerseries expansion of the log ( x ) {\displaystyle \log(x)} and the usual definition of composition of formalpowerseries. Then we have...
convolution power rely on being able to define the analog of analytic functions as formalpowerseries with powers replaced instead by the convolution power. Thus...
{\displaystyle f,g\in R} . The formalpowerseries ring]] R[[X]] also has a substitution operation, but it is only defined if the series g being substituted has...
In mathematics, the Bell series is a formalpowerseries used to study properties of arithmetical functions. Bell series were introduced and developed...
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