# Questions tagged [formal-power-series]

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223 questions
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### Is this a maximal ideal of the ring of formal power series?

Let $k$ be an algebraically closed field, and $k[[T]]$ be the ring of formal power series in variables $(T_{1},\dots,T_{n}) = T.$ Let $\mathfrak{m}^{l}$ be the ideal of $k[[T]]$ consisting ...
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### Find the radius of convergence of summesion (n=0 to ∞)(an*x^n), where an=[sin(n!)/n!] and ao=0? [duplicate]

found a question like this and there are four options R>=1 R>=2*pi R<=4*pi R<=pi, where R denotes radius of convergence. I've tried hard but cant get it how to show in equality. Also there is ...
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### Power series: convergence on right-open interval

Given the power series $f(x) = \sum_{k\geq 1} a_k x^k$ where $a_k \geq 0$, is the following statement true? Let $f(x) = \sum_{k\geq 1} a_k x^k$ with $a_k \geq 0$. If $f(x) < \infty$ for some ...
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### Proving that the given element is irreducible but not prime

This question is from Introduction to Singularities and Deformations, Greuel et al. Let $R=\mathbf{C}\langle x,y,z\rangle/\langle x^2-yz\rangle,$ where $\mathbf{C}\langle x,y,z\rangle$ is the ring ...
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### Spectrum of the ring of formal power series over integers [closed]

Let $\mathbb{Z}[[X]]$ be the formal power series ring over $\mathbb{Z}$. I want to understand the set of prime idelas $\rm{Spec}(\mathbb{Z}[[X]])$, maximal ideals $\rm{Spm}(\mathbb{Z}[[X]])$ and ...
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### Transferring big Witt vectors from “power series form” to “vector form”

I am currently attempting to read Hazelwinkel's paper "Witt Vectors 1", in which the ring of big Witt vectors $W(R)$ for $R$ a commutative unital ring is defined to be the set $R^\mathbb{N}$ with ring ...
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### On $A$ algebra homomorphisms $A[[X_1,…,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $(A,\mathfrak m)$ be a complete Discrete Valuation Ring (complete w.r.t. the $\mathfrak m$-adic topology) with fraction field $K$. Let $\phi : A[[X_1,...,X_n]]\to K$ be an $A$-algebra ...
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### On integral closedness of formal power series ring over an integrally closed domain satisfying Krull intersection principle

Let $R$ be a normal domain (i.e. an integral domain integrally closed in its fraction field) such that for every non-unit $t\in R$, $\cap_{n\ge 1} (t^n)=(0)$ ; then is it true that $R[[X]]$ is normal (...
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### How can one prove that for a formal power series f with ord(f) = 1, then ones has the following…?

Let f(t) be a formal power series and ord(f) = 1, then prove that: Res(f'(t)/f^n(t)) = 0, if n!= 1 = 1, if n=1 Maybe I don't understand what the order of a formal power series is ...
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### Preimage of a prime ideal in a ring of formal power series.

Let $k = \overline{k}$ be a field. We have the inclusion $\iota:k[x,y] \to k[\![x,y]\!]$ and the prime ideal $\mathfrak{p} = (y - \sum_{n\geq 1} x^n/n!)$. This ideal is prime, because it is the kernel ...
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### Exponentiate generating functions as formal power series

In my discrete math class, we are studying generating functions. We learned that $$e^x = \sum_{i = 0}^{\infty} \frac{x^i}{i!},$$ which is certainly an identity in calculus. However, in the ring of ...
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### Rings of formal power series is finitely generated as module.

The question rises when I was reading Kemper's "A course in Commutative Algebra." Let $K$ be a field and consider the ring of formal power series $K[[x]]$. In an exercise I proved $K[[x]]$ is not ...
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### When do we know a quotient of the polynomial ring over a local ring is torsion free

Let $R$ be a local ring (e.g. the discrete valuation ring $\mathbb C[[T]]$) and $\mathfrak{m}$ its maximal ideal. Consider the polynomial ring $R[X_1,\dots, X_n]$ in $n$ variables and a finitely ...
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### Holonomic functions and degree bounds

Let $A(x)$ is a generating function annihilated by the following differential equation of order $r$ and degree $d$. The degree of the differential equation is given by the maximal degree of the ...
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### identity of a formal power series

Recently I encounter an identity $$z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})=z_{1}^{-1}\delta(\frac{z_{0}+z_{2}}{z_{1}})$$ where $\delta(x)=\Sigma_{n\in \mathbb{N}}x^{n}$. I tried to expand both ...
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### Formal power series: $F_i(x)$ converges if and only if $\lim_{i\to\infty}deg(F_{i+1}(x)- F_i(x))=\infty.$

The following notations and definitions are taken from Richard Stanley's book Enumerative Combinatorics Volume $1,$ second edition. Recall that a formal power series $F(x)$ is of the form \sum_{n\...
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### Power rule of exponents for formal power series

I am trying to prove the following basic fact about exponentiation for a version of binomial coefficients for formal power series, based on R.P. Stanley's definition in "Enumerative Combinatorics ...