Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
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236 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
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136 views

Approximate $\ln2$ using the Maclaurin series expansion of $\ln\cos x$

Until now I have that $\ln\cos x\approx-\dfrac{x^2}{2}-\dfrac{x^4}{12}$. Since $\cos^{-1}2$ does not exist I do not know what value of $x$ to take. I suppose I need to play around with the original ...
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114 views

Taylor expansion of $x^{1/x}$

I am new to Taylor expansions and I would like to calculate the Taylor polynomial of the function $x^{1/x}=e^{(1/x)\log x}$. Since the function is not defined at $x=0$, how should I choose the point ...
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45 views

Why does the taylor expansion of a nonlinear system of differential equations exist if it has continuous second order partial derivatives?

My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$ The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever ...
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122 views

Finding closed-form approximations of the solutions of $f(x,y)=0$

Consider $$f(x,y)=\sum_{i=1}^n\dfrac{\sin(\omega_ix)}{\sin(\omega_iy)}r_i$$ where $n,\omega_i,r_i>0$ are known parameters. Restrict to domains where $ \sin(\omega_iy)\neq 0$, and by symmetry ...
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89 views

Original proof of Taylor's theorem

There are numerous proofs for Taylor's theorem, but What's the original proof for Taylor's theorem (by Taylor?)? In Wikipedia it says: Taylor's theorem is named after the mathematician Brook ...
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76 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
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86 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
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519 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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65 views

Asymptotic expansion for the solution of linear KDV eq.

Hi, The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the ...
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175 views

Proof that $\oint_r d(x,N + n) < 0 $?

Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$. And also $0 < D^M f(0) < D^{M-1} f(0)$. Let $0<T<1$ and $n$ a ...
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87 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
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140 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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258 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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260 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
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297 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
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102 views

Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
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91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
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51 views

Am I missing anything when doing this taylor expansion?

I'll make my question short, I am encountering a error when doing expansion. I am expanding $f(x)=2x^3+4x+1$ and after the expansion, things don't match. Here's what I'm doing. Let $a=5$ ...
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574 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
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46 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that ...
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37 views

Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
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106 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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38 views

Theoretical Question regarding Taylor Expansion

I received the following question during a Calculus $2.0$ course in my university. I am not a native speaker, so please excuse my English. The question is as follows: Let $f$ be a function with ...
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82 views

For which algebras Taylor series and the Fourier series can be generalized?

I'm not a professional mathematician. The question is in the title. But most of all I'd like to know about this for quaternions algebra with non commutative multiplication. I'd like to know about ...
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65 views

Taylor series of $f(x) = \arctan(x)$ converges to $\arctan(x)$

I have to find out the Taylor series of $f(x) = \arctan(x)$ and prove that it converges to $f(x)$ for any $x \in (-1, 1) $. So far I determined the Taylor series to $T_f(x) = \sum ...
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Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
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51 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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How to express $e^{yS^2}(f(x))$ in closed form where $\frac{d}{dx}=S$

$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$ is a operator $$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
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213 views

Taylor polynomial, Peano form of the remainder of f(x) and its asymptote

Could someone help me complete this or check if my reasoning so far is correct? I'm stuck at finding the oblique asymptote: Write the Taylor polynomial and the Peano form of the remainder of $f(x) = ...
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103 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
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268 views

Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
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446 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
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213 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
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54 views

Characterization of functions with fractional expansion near zero

I would like to understand if it is possible to completely characterize real-valued functions with an expansion of this type: $f(x)=f'(0)\cdot x + o(x^{\alpha})\qquad \alpha \in (1,2)$ I am not ...
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342 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$ Obviously, integer derivatives can be defined as $\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty ...
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51 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
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90 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
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279 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...
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637 views

Proving Lagrange's Remainder of the Taylor Series

My text, as many others, asserts that the proof of Lagrange's remainder is similar to that of the Mean-Value Theorem. To prove the Mean-Vale Theorem, suppose that f is differentiable over $(a, b)$ ...
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200 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
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79 views

Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for ...
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333 views

domain of convergence of a multivariable taylor series

consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
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239 views

$4$- vector Taylor expansion, sign confusion

I've been presented with a function expansion which I'm told is correct but I can't figure out where the sign in the second term might be coming from. $$ e^{i\alpha(x_\mu + \epsilon \, n_\mu)} = ...
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Visual explanation of taylor polynomials

I've just been trying to understand taylor polynomials more intuitively from a visual perspective. But as soon as the terms start using second derivatives, it becomes unclear. 1, Some thought's I've ...
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46 views

A Taylor Expansion before Taylor

Taylor expansion was introduced in its currently well known form by Brook Taylor. Though the concept as this page says, has been formulated by James Gregory. Among his other works, Gregory established ...
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18 views

higher-order (3+) Taylor expansion of a likelihood function

I was wondering what is the effect if I replace the second derivative of the log-likelihood ("Likelihood" hereafter) function with its expectation in a higher-order Taylor expansion of the likelihood ...