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Questions tagged [taylor-expansion]

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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Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ converge?

I am shockingly terrible at determining whether or not infinite series converge or not... I'm stuck on the problem: Does the series $\sum_{k=1}^{\infty} (\sin \frac{1}{k} - \arctan\frac{1}{k})$ ...
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24 views

how to show that $ \ln (x^2 e^{-x(a-1)}+1) = O(x^2e^{-x(a-1)}) $

I found this strange result from a solved exercise: (with $x\to\infty$ and $a>1$) $$ \ln (x^2 e^{-x(a-1)}+1) = O(x^2e^{-x(a-1)}) $$ I cannot explain how to get this, I can only think of using ...
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2answers
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Do I need to use McLaurin to solve the integral $ \int \frac{e^{3 x}}{x^3} dx$. [duplicate]

I need to compute this primitive : $$ \int \frac{e^{3 x}}{x^3} dx$$ but I don't know how to proceed. I tried an integration by part and get $$ \int \frac{e^{3x}}{x} dx$$ but then I am stuck... ...
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62 views

What is the series expansion of the $n$-th derivative of this : $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\text{erf}(x)}dx$

$\newcommand{\erf}{\operatorname{erf}}$ The computation of $\frac{d^n}{dx^n}\int{(e^{-x²})}^{\erf(x)}dx$ with wolfram alpha we have for $n=1, n=2, ..n=4$ interesting expansion which seems present ...
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Understanding Taylor Approximations

I am curious about what quantity a Taylor approximation actually optimizes, when it produces, as they say, the "best" possible nth-degree approximation of a function around the given x-value. ...
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50 views

The Taylor series of $\frac{1}{1-x}$ about $a=3$?

I'm having difficulty calculating the Taylor series of $\frac{1}{1-x}$ about $a=3$, and was wondering if anyone on here could help me out. Here's what I've tried so far: Attempt 1 - Take $y=x-3$ and ...
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Multidimensional complex Taylor expansion and real and imaginary part separation

I am dealing with a Lagrangian that depends on 6 complex fields plus their complex conjugate, hence, in total 12 complex fields. The Lagrangian is very long and I do not think it is necessary to show ...
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1answer
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Taylor in two variables: Can we know that two functions are different.

Consider the following setup: Two functions $f,g:\mathbb{R}^2\to\mathbb{R}$ that are twice continuously differentiable such that: $f(0,0)=g(0,0)$ $f_x(0,0)=g_x(0,0)$ $f_y(0,0)\neq g_y(0,0)$ Can we ...
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Grasping the concept of Taylor Remainder Theorem

I am trying to grasp the idea of Taylor Remainder Theorem. I want to know the way I understand it is right or wrong. Like in the linear approximation it is $$f(x) \approx f(c) + f'(c)(x-c)$$ or $$...
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Similar uses of Taylor Series and Fourier Series

Fourier series approximate given functions using sums of "sinx" curves with differing frequency. Taylor series approximate given functions using sums of power functions with differing degree. ...
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45 views

Find the linear taylor expansion for $f(x,y) = x\cos y + y\sin x$ and give a bound to the error

Consider the function $f(x,y) = x\cos y + y\sin x$. Find the linear approximation of $f$ around $(0,0)$. Find a bounding factor for the error in the region $[-1,1]\times[-1,1]$. I think by linear ...
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How to compute the summation of f(x) when the limits are in terms of y?

This is the relevant part of the question: $\sum_{y=0}^3 (x^2)$ I know the answer is $4x^2$ but I'm not certain why that is. I'm assuming that it is because there are four elements (i.e. 0,1,2 &...
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N-th Taylor Polynomial

Let $f$ and $f'$, $f''$, . . . , $f^{(n)}$ be continuous in a closed interval [$a, b$] containing a point $c$. Write down the $n$-th Taylor polynomial of $f(x)$ around $x = c$. So I know that the ...
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2answers
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How do you prove or disprove a 1:k correspondance between two series that converge to the same thing?

Suppose I have two completely different series representations of a function that can't be conventionally manipulated into each other, but converge to the same function none-the-less, like a ...
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1answer
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Find the Taylor series of argtanh(x) using sinh(x) and cosh(x)

I just finished my exam a few hours a go, and there was 1 question I couldn't answer. I was asked to derive the Taylor series of $\arg\tanh(x)$ using the fact that $$\tanh(x)=\frac{\sinh(x)}{\cosh(x)},...
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2answers
61 views

Approximate $\log(7)+\cos(1)$ with an error of less than $10^{-4}$

Evaluate $\log(7)+\cos(1)$ with an error of less than $10^{-4}$ Obviously the aim is to use Taylor's expansion with Lagrange's remainder, but where to center it? I was thinking in $e^2$, which seems ...
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3answers
66 views

Find the limit of $x(x + 1 - \sin(\frac{1}{1+x})^{-1})$ as $x \rightarrow \infty$

As the title states, I need to find the limit for $x\left(x + 1 - \frac{1}{\sin(\frac{1}{1+x})}\right)$ as $x \rightarrow \infty$, as part of a larger proof I am working on. I believe the answer is 0....
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Showing $\pi$ is irrational using taylor's theorem

To prove the irrationality of $$e = \sum ^\infty _ {n=0} \frac{1}{n!}$$ we can show that $e \lt 3$ by using a suitable geometric series. By Taylor's theorem (applied to $a=0$ and $b=1$) we know that, ...
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1answer
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Taylor series for tan x about the point 0

Write down the Taylor series for tan x about the point 0. Also write down a precise remainder term Rn(x). finding this much harder than the Taylor series for cos x, arctan x about the point 0. stuck ...
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Using taylor expansion to show the negative of the gradient gives the most rapid decrease

This is an explanation of why $-\nabla f_k$ is the better direction to step down. I undrstand why $-\nabla f_k$ is the direction that minimizes $p^t\nabla f_k$, why minimizing $p^T\nabla f_k$ will ...
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Convergence of $\sum\ln(n\sin\frac{1}{n})$ [duplicate]

How to prove that $\sum\ln(n\sin\frac{1}{n})$ converges rigorously? The idea is to expand $\sin(t)$ and $\ln(1+t)$. But I'm not sure how to do it rigorously. We have $n\sin(1/n)=n(1/n+O(1/n^3))=1+nO(...
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2answers
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Deriving the Taylor expansion $f(x+p) = f(x) + \nabla f(x+tp)^Tp$

I'm trying to derive the Taylor formula: $$f(x+p) = f(x) + \nabla f(x+tp)^Tp$$ For that I think tha I just need to use the formula for one variable taylor expansion and follow like here: https://...
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Finite differences and Taylor error

Im trying to do some numerical analysis using finite differences method, but have some questions: There is an analitical solution $u(x,t)$ and the numerical estimation $U_n(x,t)$. I want to compare ...
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1answer
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I wanted to know how to solve this question. [closed]

I had a doubt with the first question. I tried solving it and I got the upper bound to be $\frac{(6n+5)}{(2n+2)!}$
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5answers
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Prove $|\log(1-x)+x|\leqslant cx^2$

Problem: Prove $|\log(1-x)+x|\leqslant cx^2$(c is a constant) holds for $|x|<\frac{1}{2}$. My proof: $\log(x-1)=-x-O(x^2)$ then $\log(x-1)+x=-x-O(x^2)+x\implies \log(x-1)+x=O(x^2)\implies|\log(x-...
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What is a simple upper bound for $\exp\left(-\frac{1}{2}(x-(2\log(1/\delta)^{1/2}))^2\right)$ given $x \ge0$ and $\delta \in (0, 1)$?

Question For $x \ge 0$ and small $\delta \in (0, 1)$, what is a "simple" good upper bound for $$u(x,\delta) := \exp\left(-\frac{1}{2}(x-(2\log(1/\delta)^{1/2}))^2\right), $$ that doesn't involve $x$ ...
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Interchanging Integral and Summation signs

So, I know there are a few functions that don't have elementary antiderivatives, and the way I approach some of these problems is by integrating the Taylor series. In the process, I usually ...
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2answers
44 views

Approximation of Sin(2) within 10^-4

I am having some trouble with this problem. So far, I am thinking to use a similar approach to finding the approximation of sin(x) per guidance of my textbook. This would get me something along the ...
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1answer
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Taylor Series Graph vs Equivalent Function Graph

I'm sure this will be an easy question for someone with more experience. I'm just learning the Taylor series and I was told that there is an equivalent Taylor series for every polynomial, cool! So I ...
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3answers
51 views

Does the Taylor expansion and approximation centered about a point become more accurate at the point as more terms are used?

I may have done a poor job of asking this question. You can use the sine function as the Taylor expansion as an example. As you add more and more terms I can see how the Taylor representation of ...
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Convergence radius for Taylor series $f(z)=e^{z^3}\sin^3(z)-\frac{1}{2}\cos^2(z)+5\sin(z)$

Find the convergence radius for the Taylor series of $f(z)=e^{z^3}\sin^3(z)-\frac{1}{2}\cos^2(z)+5\sin(z)$. Since I have no singularities, does the Taylor series converges over all of $\mathbb{C}$, i....
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4answers
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Convergence radios for taylor series for the function $f(z)=\frac{1}{1+e^z}$ at $z=0$

I'm trying to find the Convergence radios for the function $f(z)=\frac{1}{1+e^z}$ using taylor series around $z=0$. this is what I've got so far: $\frac{1}{1+z} = \sum_{n=0}^{\infty}z^n$ thus I get: ...
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26 views

Understanding the need of regularity conditions for the Taylor expansion

Assume that the function $f(t,x) \in C^{3,4}(\Omega)$, i.e. three times continuously differentiable as a function of $t$ and four times continuously differentiable as a function of $x$. Why does the ...
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In what manner is the Taylor expansion used in the principle of stationary action?

I have a little familiarity with some basic functions that can be expressed as Taylor series but I am stuck on the concept of using Taylor expansions on the "derivation" the principle of stationary ...
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1answer
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Expansion for $\cos$ at $n\theta = C + \frac{C^3}{24}h^2 + \frac{3C^5}{640}h^4 + o(h^4)$ as a polynomial in $h$

We have $C>0, h>0$ fixed constants. And $\cos \theta = 1-\frac{h^2}{2}C^2$, for $\theta \in [0,\pi]$. Using Taylor series expansion, $$\theta = Ch + \frac{C^3}{24}h^3 + \frac{3C^5}{640}h^5 + o(...
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1answer
81 views

How to approximate the solution of $y=a \arctan(x/a)-\arctan (x)$

After this question, related to the Prandtl–Meyer function $$\nu(M) = \int \frac{\sqrt{M^2-1}}{1+\frac{\gamma -1}{2}M^2}\frac{\,dM}{M}= \sqrt{\frac{\gamma + 1}{\gamma -1}} \tan^{-1}\left( \sqrt{\...
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3answers
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Differential Polynomials(?)

Consider an equation of the form: cy"+cy'+cy Or something of the form. Essentially, it's a polynomial but instead of powers, there are derivatives. Do these kind of things have a name? Or are they ...
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Simple lower bound on Gaussian CDF evaluated at sum: $G(s + t)$ in terms of $G(s)$, with $s, t \ge 0$ and $s \le 1$

Let $G: s \mapsto \int_{-\infty}^s g(s)ds$ be the CDF of the standard Gaussian (with $g(s) := (2\pi)^{-1/2}\exp(-s^2/2)$ the density) and $s \le 0 \le t$. Question what is a simple lower bound for ...
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Taylor expansion of a Neural Network

Let $f(\mathbf{x})$ represent a trained neural network with ReLU() activation functions and input $\mathbf{x} \in \mathbb{R}^d$ ($\mathbf{x}$ could be, for example, an image with dimensionality $d$). ...
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207 views

What uniquely characterizes the germ of a smooth function?

Let $X$ be the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which are infinitely differentiable at $0$. Let us define an equivalence relation $\sim$ on $X$ by saying that $f\sim g$ if ...
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Bound the residual between function and Taylor approximation

Taylor approximation of function f(•) can be written as follows: $$f(x+h)=F_{x,p}(x+h) + o(||h||^{p})$$ Assume that f is convex and p times differentiable on $dom / f$. Denote by $L_p$ uniform bound ...
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2answers
85 views

Series expansion of $\tan^2$ and $\tanh^2$

Are there known closed formula expressions for their power series expansion at the origin? I couldn't find anything online. Edit: (to clarify) Of course we could simply take the series expansion of ...
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Why ignoring higher order terms in Taylor's series won't cause a problem?

Please don't mark this as a duplicate question as I have read similar questions on SE but couldn't find my answer. I have been studying numerical methods and an important part of the introduction to ...
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1answer
37 views

Finding simple fraction decomposition with help of Taylor's Theorem and Residues theorem

I have this problem: $F(s)=\frac{a_{1}s+a_{2}}{s^{2}(s^{2}+12s+37)}$ I thought in Taylor's expansion of a function f(s) in s=0: $f(s)= f(0)+f'(0)\cdot s+f''(0)\cdot s^{2}+\cdots$ and then I defined:...
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31 views

References for the “Principal part method”?

Here is the statement : If $f$ is a continuous function on $\mathbb{R}$ and there exists $p\in \mathbb{N}^*$ and a real number $\gamma>0$ such that $f(x)\underset{x\to 0}=x-\gamma x^{p+1}+ o(x^{...
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1answer
31 views

Finding Laurent series of a function by changing variable

I'm trying to find Laurent series of the following function at $$\mid z\mid<1$$ My function is as follows: $$f(z)=\dfrac{1}{z(z-1)(z-2)}=\dfrac{1}{z}\left(\dfrac{1}{(z-2)}+\dfrac{1}{(1-z)}\right)$$ ...
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0answers
26 views

Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that I do not mean multilinear PCA, which operates on data tensors, but some form of SVD which can produce, say, ...
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3answers
41 views

Is $|\ln(1+t) - t| \leq t^2$ for $|t| \leq \frac{1}{2}$ obvious?

In a textbook of probablity and statistics there is a use of approximation by Taylor series that $\ln(1 + t) \doteq t$. It is stated that that the accuracy of the approximation is due to $|\ln(1+t) - ...
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1answer
68 views

Stability of Hölder continuity under nonlinear maps

Let $f, g \in C^{\alpha}(\mathbb{R}^d, \mathbb{R})$ be Hölder continuous of degree $\alpha$ and let $F \in C^2_b$ be a bounded, twice continuously differentiable function with bounded derivatives. I ...
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29 views

Understanding Lagrange error bound

Estimating $\ln(0.9)$ using a Taylor polynomial about $x=1$ what is the least degree of the polynomial that assures an error smaller than $0.0010$? (The $n^{th}$ derivative of $(-1)^{n-1}\cfrac{(n-1!)...