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Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
Every field is a Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.) Complete Hausdorff local rings, such as the ring of p-adic integers and rings of formal power series over a field, are Henselian. The rings of convergent power series over the real or complex numbers are Henselian.
Quotient rings of the ring are used in the study of a formal algebraic space as well as rigid analysis, the latter over non-archimedean complete fields. Over a discrete topological ring, the ring of restricted power series coincides with a polynomial ring; thus, in this sense, the notion of "restricted power series" is a generalization of a ...
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 the complex case, algebraic geometry begins by defining the complex affine space to be . For each , we define , the ring of analytic functions on to be the ring of holomorphic functions, i.e. functions on that can be written as a convergent power series in a neighborhood of each point.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
3. An analytic ring is a quotient of a ring of convergent power series in a finite number of variables over a field with a valuation. analytically This often refers to properties of the completion of a local ring; cf. #formally 1. A local ring is called analytically normal if its completion is an integrally closed domain. 2.
The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation.