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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172 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of ...
6
votes
0answers
94 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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92 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 ...
6
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91 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 $x_{0}$ ...
5
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64 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 ...
4
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48 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 ...
4
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93 views
+50

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. Also, what are the actual ...
4
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40 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 ...
4
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238 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
166 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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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
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0answers
43 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
394 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
40 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
47 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
74 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
votes
0answers
85 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 $$ ...
3
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330 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
151 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
53 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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46 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
votes
0answers
77 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
266 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
68 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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277 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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40 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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0answers
41 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
2
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0answers
30 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
13 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
54 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 ...
2
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0answers
70 views
+50

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 ...
2
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0answers
79 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 ...
2
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0answers
68 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) = ...
2
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30 views

Entire periodic $f(z)$ with more than 50 % of the derivatives $0$?

Im looking for a real-entire function $f(z)$ such that for any complex $z$ : $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number. $2)$ $f(z)= 0 + a_1 z + a_2 z^2 + a_3 z^3 + ...$ where more than ...
2
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0answers
20 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
2
votes
0answers
31 views

Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
2
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0answers
54 views

Wynn-epsilon convergence

How could I use the Wynn-epsilon alghoritm in Matlab to accelerate the convergence of a Maclaurin series? I want to extimate the first derivative of $f(x)$, so $$f'(x)= \sum_{k=0}^\infty ...
2
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0answers
36 views

Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
2
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0answers
41 views

What is the 2nd order taylor polynomial of f(x,y)?

I'm just computing the 2nd order taylor polynomial for $f(x,y) = tan(x + 3y + \frac{\pi}{4})$ centered at (3,-1) and wondering if I have done this correctly or if anyone has any suggestions on how I ...
2
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0answers
372 views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
2
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0answers
81 views

Inequality sine power series

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
2
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0answers
78 views

Taylor series $\frac{\sin x}{x}$ convergence

I needed the Taylor series for $f(x) = \frac{\sin x}{x}$ in $a = 0$. I started with $ f(x) = \frac{1}{x} \cdot \sin(x) $, used the existing $sin$ Taylor series and multiplied by $\frac{1}{x}$: $$ ...
2
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0answers
455 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
2
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0answers
240 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
2
votes
0answers
35 views

Optimization, descent direction, neccessary condition

I'm learning about nonlinear, unconstrained optimization. In my book it says that a descent direction $p_k$ must satisfy: $$p_k\nabla f(x_k)^T < 0$$ This seems to mean that $p_k$ must be obtuse to ...
2
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0answers
179 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 ...
2
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0answers
57 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
2
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0answers
69 views

Taylor expansion proof

It's pretty clear to me that in this expansion: $$p(x) = f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+f'''(0)\frac{x^3}{3!}+\cdots$$ When I assume $x=0$, $p(0)$ is gonna be equals to $f(0)$ and all its ...
2
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0answers
1k views

Taylor's Theorem for Multivariate Functions

Please look at this theorem in Wiki regarding Taylor's theorem generalized to multivariate functions: Multivariate version of Taylor's Theorem The version stated there is one that I'm not familiar ...
2
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0answers
254 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 $(a, b)$ ...