The taylor-series tag has no usage guidance.

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### aproximate sum involving binomial coefficients

I have the problem for computing the j-derivative of a logarithm, with $j\gg1$
\begin{equation}
c_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|_{s=0},
\end{equation}
...

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159 views

### Lower Bounds for the Roots of Polynomials

I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function.
For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are
...

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**1**answer

51 views

### Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let ...

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109 views

### Zeros of partial sums of $(1+z)/(1-z)$ near $z=-1$

Let $p_n(z)$ be the $n^\text{th}$ partial sum of the Maclaurin series for $f(z) = (1+z)/(1-z)$. For large $n$ the zeros of $p_n$ appear to avoid the point $z=-1$:
Figure: Zeros of $p_{40}$ and the ...

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145 views

### Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
...

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125 views

### Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem:
Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...

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**1**answer

239 views

### A function with positive decreasing Taylor coefficients?

The function $\frac{x}{\ln(1-x)}$ has a Taylor series $-1+c_1x+c_2x^2+\cdots$ and I want to show $c_1>c_2>\cdots>0$. More generally, is there a result about how a function has positive Taylor ...

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113 views

### Determing signs of Taylor coefficients in entire functions

This is a continuation of Determining when combinatorial sums are zero
Suppose $f(x)$ is an entire function approximated by polynomials with only negative real zeros. Suppose further that ...

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100 views

### why can't taylor series capture memory effects? [closed]

I am trying to understand when to use Volterra series.I found this on wikipedia 'The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series ...

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**1**answer

124 views

### Discrete Taylor's Formula in n dimensions [closed]

I am searching for discrete form of Taylor's formula in n dimensions. Please share the appropriate resources.

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102 views

### Properties and name of some polynomials

I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...

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349 views

### An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this:
$G(s) = e^{a(s-1)^2}=\sum s^np(n)$
I need first to do Maclaurin expansion of the exponential and ...

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96 views

### Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you.
This post is related to zeros of partial sum of Taylor series of $e^x-1$.
Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...

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102 views

### Positive, Uni-modal, Log-concave Combinatorics

We define a sequence, $\{a_n\}_{n=0}^\infty$, to be a uni-modal sequence if for some $m$, $$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$
We define a sequence, ...

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333 views

### An apparently simple question (behaviour at infinity of a power series)

Let $(a_n)$ be a sequence of real numbers, and suppose that the real power series (function) $S(x):=\sum_{n=0}^{\infty} a_n x^n$ converges for every $x\in\mathbb{R}$.
$\mathbf{Question}$: Suppose ...

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**1**answer

75 views

### An algebraic equation question [closed]

My question is this:
If $\frac{\sqrt[n]{\prod_{i=1}^n(p_i + 1)}}{\sqrt[n]{\prod_{i=1}^n(m_i + 1)}} = e ^\beta$
can I find an expression (either exact or approximate) for ...

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**1**answer

158 views

### Approximate the square root of (1-X) efficiently through (nested) products

Currently, I encountered a problem of approximating the following
series:
$$
(I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots
$$
where ...

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**0**answers

189 views

### derivatives of composite function [closed]

There's a formula for the $n$th derivative of a composite function $f(g(x))$ - it's called Faa di Bruno's formula - but I'm not really interested in the formula but in the proof given in the book of ...

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552 views

### Taylor series coefficients

This question arose in connection with A hard integral identity on MATH.SE.
Let
$$f(x)=\arctan{\left (\frac{S(x)}{\pi+S(x)}\right)}$$
with $S(x)=\operatorname{arctanh} x -\arctan x$, and let
...

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**1**answer

143 views

### A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) ...

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231 views

### Laurent expansion of inverse of vandermonde determinant

I wish to find the coefficients of the Laurent expansion of the inverse of the Vandermonde determinant, that is, the Laurent expansion at 0 of
$$\prod_{1\leq i<j \leq n}(x_j-x_i)^{-1}.$$
We can ...

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**1**answer

574 views

### construct a power series with infinitely many zeros in the complex plane, bounded coefficients???

Hi all.
I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...

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**1**answer

123 views

### Estimate the scale of the power series with Poisson pdf/pmf-like terms

I would like to have an estimate for the series
$$P(t) = \sum\limits_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$
where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, ...

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489 views

### References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
...

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133 views

### Series expansion with remaining $log n$

Hi,
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a ...

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**1**answer

206 views

### Inequality of Partial Taylor Series

Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} ...

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228 views

### A question about approximation of Real analytic functions

Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$
for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in ...

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417 views

### Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative?
Is there a simple relation between the entropy of a matrix and its characteristic polynomial?

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179 views

### Taylor expansion convergence relation to power-spectrum

Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty ...

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222 views

### nonlinear delay differential equation

Consider the delay differential equation:
$ y_x(x) = \sqrt{y(x-\bar{x})} $
where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter.
This equation does not seem to have a ...

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3k views

### Numerical Computation of arcsin and arctan for real numbers [closed]

I'm coding some numerical methods and I do not know what the correct analysis would be for choosing the implementation for $arcsin$ and $arctan$ for real numbers. Here's what I know:
Both functions ...

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361 views

### Converse of the taylor's theorem in Banach Spaces

I would like to known if the following converse of the taylor's theorem is true:
Let $E$, $F$ Banach spaces, and $f:E\rightarrow F$ continuous. Suppose there are $k$ continuous functions $T_i: E ...

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**1**answer

401 views

### Approximation:- Algorithmic considerations

Hello
I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...

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329 views

### Applying the ideas of power series to certain convolutions - which identities transfer?

Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...

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856 views

### Taylor's series for Lie groups

Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) ...

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**1**answer

283 views

### method for getting function from power series/perturbation series

is there any definite method or algorithm,software to get exact function or expression from series.e.g we get series solution of differential equation and we want exact expression rather than ...

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**1**answer

387 views

### Higher order Approximation of Lie groups [closed]

Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its ...

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**0**answers

240 views

### Jet spaces for maps with constraints

Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps:
Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...

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### Integral representation of higher order derivatives

I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I ...

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**1**answer

497 views

### Series approximation(s) of a difficult recursive equation

New user here. I'm working on trying to get asymptotic solutions to the following recursive function:
$f(r)=\frac{1}{r-k}\lgroup\sqrt{\frac{2}{k^2-1}}+\sqrt{\frac{1}{2k^2-1}}\rgroup$ (Eqn. 1)
...

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299 views

### about power series for iterated logarithms

The question is motivated by this one. It turned out (see my comment there) that the coefficients of the Taylor series for $\log\log x$ at $x=e$ have nice combinatorial description from Sloane's ...

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### power series of the reciprocal… does a recursive formula exist for the coefficients. [closed]

Hello
If $f(x)=\sum _{n=0}^{\infty } b_nx^n$, and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reprical of f can be written down. The first few terms are:
$d_0 = ...

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**1**answer

823 views

### Padé approximation - usability in iterative algorithms

Firstly, I have to say that I don't understand Padé approximation well.
But I discovered that, it is more precise than Taylor series.
I have to create approximation for these functions: Log(x) and ...

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**1**answer

516 views

### Can Convergence Radii of Padé Approximants Always Be Made Infinite?

I've found (as have others), that for some analytic functions, a Padé approximant of it has an infinite convergence radius, whereas its associated Taylor series has a finite convergence radius. ...

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3k views

### Why is $ \frac{\pi^2}{12}=\ln(2)$ not true?

This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.
Let us begin with (one of the many) equalities established by Euler:
...

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2k views

### Estimate for tail of power series of exponential function?

I would like to have an estimate for the infinite series
$$
\sum_{k=B}^\infty \frac{A^k}{k!},
$$
where $A$ is a large positive quantity and $B$ is just a little bit bigger than $A$, namely, $B = A + C ...

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2k views

### Using Weierstrass’s Factorization Theorem

I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} ...

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995 views

### What characterizes rational functions with nonnegative integer Taylor coefficients?

I believe that there is a statement along the following lines (I would, of course, love to be corrected): a formal power series is the Taylor expansion of a rational function if and only if the ...

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1k views

### Taylor series of a complex function that is not holomorphic

I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.
Bonus question: Can I ...

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4k views

### Roots of truncations of $ e^x - 1$

During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...