Questions tagged [taylor-series]

Taylor series is a method to analyze functions as polynomials.

Filter by
Sorted by
Tagged with
73 votes
7 answers
7k 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 ...
Vectornaut's user avatar
  • 2,214
29 votes
2 answers
16k views

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

Let $f(x)=\sum _{n=0}^{\infty } b_nx^n$ and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are: $d_0 = \frac{...
AUK1939's user avatar
  • 569
25 votes
4 answers
5k 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: ...
Max Muller's user avatar
  • 4,435
25 votes
3 answers
15k 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)?
Eric Wilson's user avatar
19 votes
2 answers
2k 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 ...
Pietro Majer's user avatar
  • 56.3k
18 votes
3 answers
1k views

A curious series related to the asymptotic behavior of the tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
Vladimir Reshetnikov's user avatar
11 votes
1 answer
378 views

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...
André Henriques's user avatar
10 votes
2 answers
1k views

Fourier series of $\log(a +b\cos(x))$?

By numerical computation it seems like, if $a_0 < a_1$: $$ \begin{multline} \log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\ + \frac{a_0}{a_1}\cos(\omega t) - \frac{...
Alister Trabattoni's user avatar
9 votes
3 answers
4k 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 ...
Greg Martin's user avatar
  • 12.7k
8 votes
3 answers
2k 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 ...
Theo Johnson-Freyd's user avatar
8 votes
1 answer
2k views

Polynomial approximation for square root function with fast convergence and bounded coefficients

Let $\delta, \varepsilon \in (0,1)$. I am interested in a sequence $\{f_n\}$ of polynomial approximations of the square root function $x \to x^{1/2}$ on $[\delta,1]$, of the form $$ f_n(x) = \sum_{i=0}...
Mark M. Wilde's user avatar
8 votes
1 answer
1k views

Taylor expansion of cumulant generating function

For the characteristic function $\mathbf E e^{i t X}$ of a random variable $X$ with $n+1$ finite moments, there is the well known and easy to prove bound on the remainder of the Taylor series $$\left\...
Julian's user avatar
  • 613
8 votes
1 answer
479 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...
mick's user avatar
  • 721
7 votes
2 answers
6k 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} \...
vonjd's user avatar
  • 5,855
7 votes
3 answers
646 views

Transformation converting power series to Bernoulli polynomial series

I wonder, can anyone describe an expression or formula of a transform that converts $$\sum_{k=0}^\infty \frac{a_k x^k}{k!}$$ into $$\sum_{k=0}^\infty \frac{a_k B_k(x)}{k!}$$ where $B_k(x)$ are ...
Anixx's user avatar
  • 9,316
7 votes
1 answer
1k views

Is there a bound for Lipschitz constant in terms of second differences?

It is easy to show that if $f\colon[0,1]\to\mathbb R$ and $|f|\leq A$ and $|f''|\leq B$ then~$|f'|\leq 4A+B$. Indeed, by Taylor formula with remainder $f(x)=f(c)+(x-c)f'(c)+\frac12(x-c)^2f''(d)$ where ...
Mikhail Katz's user avatar
  • 15.1k
7 votes
1 answer
657 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 ...
Bo WANG's user avatar
  • 71
7 votes
0 answers
287 views

Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
Jannik Pitt's user avatar
  • 1,103
6 votes
2 answers
1k views

Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable

This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function ...
Alex's user avatar
  • 210
6 votes
2 answers
396 views

What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$?

$\delta$ is a positive number. Is this Taylor expansion of some function?
zhouch2012's user avatar
6 votes
1 answer
309 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 ...
Per Alexandersson's user avatar
6 votes
2 answers
204 views

Taylor expansion theorem for Gateaux differentiable functions

I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet ...
Antonio Martins Alves Veloso d's user avatar
6 votes
0 answers
2k views

Do smooth cutoff functions analytically continue functions?

My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
Caleb Briggs's user avatar
  • 1,662
5 votes
2 answers
421 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 ...
doubllle's user avatar
  • 153
5 votes
1 answer
353 views

Bounding higher derivatives of $f(x) = 1/(1+x^2)^r$

Let $r\in \lbrack 0,\infty)$. Define $f(x) = 1/(1+x^2)^r$. It would seem to be the case that $$|f^{(k)}(x)|\leq \frac{2r \cdot (2r+1) \dotsb (2r + k-1)}{(1+x^2)^{r + k/2}}$$ for all even $k\geq 0$. ...
H A Helfgott's user avatar
  • 19.3k
5 votes
1 answer
349 views

Family of functions with prescribed derivatives

Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
Ali's user avatar
  • 4,045
5 votes
2 answers
600 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 $$\...
Fly by Night's user avatar
5 votes
2 answers
502 views

Taylor $k$-differentiability of a real function at a point

I am interested in the standard name for the following weak form of $k$-differentiability. Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
Taras Banakh's user avatar
  • 40.7k
5 votes
3 answers
735 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 $$f(x)=\...
Zurab Silagadze's user avatar
5 votes
1 answer
327 views

Asymptotic growth of the of Taylor coefficients of the inverse of a function

Let $f(x)=\sum_{n\geq 1} c_n\cdot x^n$ be a function given by a power series. Further there is some $\alpha >1$ such that for all $n$, $c_n = \Theta(1/n^{\alpha})$. What can one say about the ...
user116726's user avatar
5 votes
1 answer
348 views

Taylor-like expansion for a holomorphic function in non-simply-connected domain

Suppose $f$ is a holomorphic function in a simply connected open set $U$, and we know it's Taylor expansion at a point $p\in U$. We can then find a holomorphic map $g$ of $U$ to the unit disc which ...
Peter Kravchuk's user avatar
5 votes
0 answers
610 views

The Basel problem revisited?

In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
mathoverflowUser's user avatar
5 votes
0 answers
253 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
Arrow's user avatar
  • 10.3k
5 votes
0 answers
74 views

Complexity of calculating $f^{(n)}(0)$/extracting a coefficient of a functions taylor-series

Many combinatorial problems can be solved using generating functions. In such a case, we obtain a function $f(x)$, which (for usual) has a taylor-expansion: $$ f(x) = \sum_{n\ge 0 } a_n x^n $$ So ...
Sudix's user avatar
  • 151
5 votes
0 answers
269 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 ...
mike's user avatar
  • 603
4 votes
1 answer
1k views

Taylor expansion of determinant of Riemannian metric in normal coordinates up to higher order

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Let $p\in M$, and let $\{x^i\}_{i=1}^n$ be normal coordinates centered around $p$. Using Jacobi field, one can show that the metric $g$ has the ...
Anonymous amateur's user avatar
4 votes
1 answer
3k 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) ...
Alessandro Saccon's user avatar
4 votes
3 answers
471 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 ...
Josh's user avatar
  • 51
4 votes
2 answers
229 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 ...
user900's user avatar
  • 41
4 votes
4 answers
629 views

What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

It is known that \begin{equation*} \tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2} \end{equation*} and \begin{equation*} \ln\tan x=\ln x+\...
qifeng618's user avatar
  • 828
4 votes
0 answers
140 views

Covergence of fractional Taylor series

Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function: $$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$ ...
Мікалас Кaрыбутоў's user avatar
4 votes
0 answers
656 views

Functional Taylor expansion for differential entropy

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is $h(f)=-\int f(x)\log f(x) dx$. In the literature of differential entropy estimation, ...
AD1984's user avatar
  • 145
3 votes
1 answer
446 views

Taylor expansion of Stieltjes Transform

I'm trying to derive a very basic result stated in several books on random matrix theory (e.g. Terry Tao's book and Potters & Bouchaud's book). Given a symmetric matrix $A \in \mathbb{R}^{N \times ...
digbyterrell's user avatar
3 votes
2 answers
530 views

Are there any techniques that can be used in the case when a Neumann series doesn't converge?

Suppose we have a bounded linear operator $A = A(\gamma):H_1\to H_2$ where $H_1$ and $H_2$ are Hilbert spaces and $\gamma>0$ is some parameter, and we are interested in the solution to $$ (I-A)x = ...
ManUtdBloke's user avatar
3 votes
4 answers
1k 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): ...
CHL's user avatar
  • 31
3 votes
2 answers
241 views

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} ...
user3209698's user avatar
3 votes
1 answer
182 views

Other expansion for positive Taylor expansion

I was thinking of the following problem. Let $f$ be a Taylor expansion and $a_k$ the associated coefficients, $$\forall x\in\mathbb{R},~f(x)\triangleq\sum_{k=0}^\infty a_kx^k.$$ Let suppose that we ...
NancyBoy's user avatar
  • 175
3 votes
1 answer
2k views

Taylor series on a Riemannian manifold

I need some help for the following problem. Let $M$ a riemannian manifold and $f$ a smooth differential function, then consider the following integral $$\int_M \Gamma(x,y)(f(y)-f(x))dV_y$$ where $dV_y$...
Pether Cll_'s user avatar
3 votes
1 answer
186 views

Probabilistic Taylor theorem for concave functions

This paper proves a probabilistic version of Taylor's theorem \begin{equation*} \mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...
Dejan Evisal's user avatar
3 votes
1 answer
515 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 ...
PatrickT's user avatar
  • 174