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1
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0answers
58 views

Identity of Bernoulli polynomials

consider the Bernoulli polynomials defined by the generating function: $$\left(\prod_{i=1}^m \frac{a_i}{\left( e^{a_i}-1 \right)}\right)e^{xt}=\sum\limits_{n=0}^{\infty}B^{m}_n\left(x\vert ...
-1
votes
0answers
51 views

Bernoulli Numbers and radius of convergence [migrated]

consider the function $f(x)=\frac{x}{e^x-1}$. Since the function $\frac{1}{f(x)}=\frac{e^x-^1}{x}=\sum\limits_{k=0}^{\infty} \frac{x^k}{(k+1)!}$ has a taylor expansion with $\frac{1}{f(0)}\neq 0$ we ...
4
votes
2answers
316 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 ...
4
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0answers
55 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 ...
2
votes
0answers
87 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, ...
3
votes
3answers
287 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 ...
2
votes
1answer
73 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 ...
3
votes
1answer
136 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 ...
3
votes
0answers
145 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 ...
6
votes
3answers
468 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 ...
1
vote
1answer
128 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) ...
6
votes
1answer
209 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 ...
0
votes
1answer
436 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 ...
1
vote
1answer
98 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 $k$, ...
1
vote
0answers
236 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 ...
0
votes
0answers
118 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 ...
0
votes
1answer
187 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} ...
-1
votes
1answer
209 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 ...
0
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0answers
234 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?
2
votes
2answers
157 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 ...
3
votes
1answer
178 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 ...
2
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2answers
2k 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 ...
3
votes
1answer
282 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 ...
2
votes
1answer
384 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 ...
1
vote
0answers
219 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 ...
2
votes
1answer
660 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) ...
1
vote
1answer
278 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 ...
0
votes
1answer
351 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 ...
0
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0answers
234 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 $M$. ...
17
votes
2answers
1k views

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 ...
2
votes
1answer
396 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) ...
28
votes
3answers
3k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Hi. Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" ...
3
votes
1answer
289 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 ...
6
votes
2answers
3k views

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 = ...
1
vote
1answer
645 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 ...
0
votes
1answer
437 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. ...
14
votes
3answers
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 afore mentioned 'identity'. Let us begin with (one of the many) equalities established by Euler: ...
7
votes
2answers
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 ...
3
votes
2answers
1k 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} ...
6
votes
3answers
854 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 ...
-2
votes
2answers
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 ...
62
votes
6answers
3k 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 ...
3
votes
4answers
518 views

Closed form of a type of generalised exponential functions

Let a generalised exponential function exp_{m,n} (I'm not sure if this notation is already used by anything else) be defined as such, for n a positive integer and m between 0 and n-1 (inclusive): ...
3
votes
2answers
183 views

Properties of signomial Functions in one variable

I am interested in functions of the form: \sum_{j=1}^\infty a_j x^{p_j} where p_j can be any non-negative real number. Wikipedia has informed me that this is a subset of the signomial functions, but ...
7
votes
3answers
3k views

What's an example of a function whose Taylor series converges to the wrong thing?

Can anyone provide an example of a real-valued function f with a convergent Taylor series that converges to a function that is not equal to f (not even locally)?