# Questions tagged [power-series]

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### A proof of an interesting inequality

If $0<\beta<1$ and $0<x<1,$ how to prove that $$h(x)-2x+(4-2^{1+\beta})x^{1+\beta}<0,$$ where $$h(x)=(1+x)^{1+\beta}-x^{1+\beta}-1.$$The numerical simulation shows that it is true.
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### In the rational numbers, is every convergent power series a Taylor series for a rational function?

David Roberts wrote in the comment section of the blog post "Convergence of an infinite sum in the rationals" the following paragraph: Someone mentioned (I think on Twitter) that the Taylor ...
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### Is there an infinite combinatorics of common transseries expansions?

By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...
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### Can we write $e^{-\alpha x}$ as $\sum_{n=0}^\infty c_n\left(\alpha\right)\gamma\left(x\right)^n$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$

Do there exist continuous functions $c_n\colon\mathbb R^+\to\mathbb R$ and $\gamma\colon\mathbb R^+\to\mathbb R$ such that $\lim_{x\to\infty}\gamma\left(x\right)=0$ and the following equation is true ...
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### How to prove this equation? [closed]

How to prove this equation: \begin{align}\sum_{k=1}^{n}\cos ^{2m+1} \frac{(2k-1)\pi}{2n+1} =\frac{1}{2}\end{align} where $k,m,n\in \mathbb{N}^*$.
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### Lagrange inversion for power-series with rational powers

One can use Lagrange inversion to find the power series $F(x)$, which solves $F(x) = x(1+F(x)^p)$, where $p$ is a positive integer. Now, what if $p$ is not an integer, but rather a positive rational ...
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### 'Lie correspondence' for formal power series in non-commuting indeterminates

This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following. Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...
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### Closed form series for reciprocal cubic function

consider a cubic of the form f(x)=$x^3-2x+z$ Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
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### Two questions about Bell polynomials

In my research in Quantum Field Theory, I have encountered two questions that involve partial Bell polynomials: Let $u$ and $x_i$ be indeterminates. I have checked that the following conjectured ...
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### Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$

I would like to construct an approximation of the product \begin{equation} f(z) = \frac{1}{\overline{z}-a} \frac{1}{z-b}, \end{equation} where $a, b \in \mathbb{C}$, and $|{a}/{z}|, |{b}/{z}| <1$. ...
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### Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?

According to numerical simulation, the relationship $$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$ where $\Gamma$ is the Gamma function seems to be true. Do you ...
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### Divergent series summation beyond natural boundaries

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
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### Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ? I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...
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### Is there a closed form of $\displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$

where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...
1 vote
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### How to explicitly obtain an analytic function whose power series coefficients are sums over integer compositions?

Starting with the following differential equation, \begin{eqnarray} x \frac{\partial^3}{\partial x^3} P[h, x] - \frac{\partial^2}{\partial h^2} \left( h \frac{\partial}{\partial h} P [h, x] \right) ...
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### How could this difference in series of power of zeros associated to counting integers and counting primes be explained?

Introduction: In this 1992 paper, J.B. Keiper (an amazing person, who tragically died way too young), derives several power series expansions of the Riemann $\xi$-function that involve infinite sums ...
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### Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series)

The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be ...
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I have been wrestling with the following problem for some time now. If possible, I would prefer a hint rather than a full solution, because I would like to "solve" it myself. Let $f(z)$ be a ...