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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27
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852 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
7
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152 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 $x_{0}$ ...
6
votes
0answers
70 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 ...
6
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0answers
130 views

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 ...
6
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0answers
110 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 ...
5
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70 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 ...
5
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0answers
66 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 ...
5
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0answers
75 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
5
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0answers
288 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 ...
4
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49 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 ...
4
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78 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 ...
4
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0answers
207 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 ...
4
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0answers
250 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} ...
4
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0answers
226 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 ...
4
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0answers
95 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 $$ ...
4
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0answers
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 ...
4
votes
0answers
48 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 $a=5$ ...
4
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0answers
489 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 ...
3
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0answers
70 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 $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
3
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0answers
30 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 ...
3
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0answers
42 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 ...
3
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0answers
57 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 ...
3
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0answers
49 views

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 ...
3
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0answers
42 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 $\exp(aD)f(x)=f(x+a)$?
3
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93 views

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 ...
3
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0answers
87 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 ...
3
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0answers
397 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 $$ ...
3
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0answers
175 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 ...
3
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0answers
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 ...
3
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0answers
273 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 ...
3
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0answers
50 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 ...
3
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0answers
89 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 ...
3
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0answers
358 views

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 ...
3
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0answers
183 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!
3
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0answers
75 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 ...
3
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0answers
295 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 ...
2
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0answers
42 views

Taylor expansion in terms of known Taylor series

Let $f(x)$ be a "nice" function with the following properties: $f$ is a real analytic, strictly increasing, odd and bounded function, i.e. $f(-x)=-f(x)$ and $-1<f(x)<1$. Further, let $f(x)>0, ...
2
votes
0answers
83 views

About sparse polynomial squares

Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$ W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\} $$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...
2
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0answers
24 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
2
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0answers
32 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
2
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0answers
31 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
2
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0answers
28 views

Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
2
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0answers
39 views

Expansion of gamma function

The lecturer wrote down $\Gamma(x-2)=-\frac{1}{2x}+\cdots$ , but I can't figure out where this comes from? It needs to be in this form so that I can cancel the $x$ with the expansion of another ...
2
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0answers
45 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = ...
2
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0answers
55 views

Can we always exploit Taylor's approximation in order to avoid using l'Hopital rule?

We are given two functions, namely: $f, g: U^\circ\subset \mathbb{R}\to \mathbb{R}$ where $0 \in \overline{U}$. Suppose you want to compute the limit $$\lim_{x \to 0} \frac{f(x)}{g(x)}$$ and that ...
2
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0answers
55 views

Asymptotic expansion of integral (Laguerre)

Consider $$L_n = \frac{1}{2\pi i } \oint_{C'} \frac{1}{(1-t)^{\alpha+1} t^{n+1}} e^{-\frac{xt}{1-t}} dt\,\,\,\,(1)$$ where $C'$ is an anticlockwise contour around zero. Now set $\alpha = n$ and I want ...
2
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84 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
2
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0answers
40 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
2
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0answers
20 views

Upper bound on derivatives of very high order?

I am doing a calculation where I am estimating a value $\omega$ by a Taylor polynomial. I know that $\omega \cdot a = f(b)$ and thus I can estimate $\omega$ by $a \cdot T_n f(b) $ where $T_n$ is the ...
2
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0answers
64 views

Book Request: Taylor's Theorem for functions $f: \Bbb R^n \to \Bbb R^m$

I'm looking for a resource (e.g. a book, website, or arxiv paper) that goes over the general case of Taylor's theorem, with a full proof and examples. Do you guys know of any material that covers ...