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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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Quadratic Taylor coefficient for free by normalizing; when is second Taylor coefficient real?

Let $f: \mathbb{R} \to \mathbb{C}$ and suppose that we know $|f(\lambda)| = 1$ for all $\lambda$. Consider the Taylor series around $0$: $$ f(\lambda) = a + b\lambda + c\lambda^2 + \cdots. $$ Instead ...
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Write out first four terms of Taylor series $f(x)= \frac{1}{(2-x)^2}$

To do it the simple way, I know $f(x)=\frac{1}{(2-x)^2}$ can be written as $$g(x)= \frac{1}{2-x}$$ and then you can differentiate term by term. What exactly does that mean?
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Compute limit of complicate function.

The question is compute $$ \lim_{x \to 0}\ \dfrac{x^2e^{x^2}+\dfrac{1}{2}\log(1+2x^2)-2x^2}{1+\arctan(x^6)-\cos(x^3)} $$ using Taylor series expansion around the origin, you should not use L'Hopital'...
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Calculating $\sum_{k=1}^\infty 2^{-k}(e^{-k}-e^{-k-1})$ [on hold]

Pretty basic question - I'm just not that experienced with calculating summations and would love help with understanding the steps involved (I computed it with Mathematica but couldn't see the steps ...
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What are some practical applications of successive differentiation?

Before starting to learn something, I always wonder whats its application. So would you please give some practical examples of application of Successive differentiation and concepts related to it such ...
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Second-Order Taylor Series Terms In Gradient Descent

My machine learning textbook states the following when discussing second-order Taylor series approximations in the context of Gradient descent: The (directional) second derivative tells us how well ...
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Discrete derivative formulation by Taylor Expansions

I'm following the paper "Cordova 2014, Comparative Study of two compact finite difference methods". It states: Given a discretization of a line by $x_j = -1 + jh$, where $j = \{ 0, 1, \ldots, N \}$ ...
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Proving:$\prod\limits_{n=1}^\infty\left(1+x^{2^{n-1}}\right)=\dfrac1{1-x}$

Proving:$\prod\limits_{n=1}^\infty\left(1+x^{2^{n-1}}\right)=\dfrac1{1-x}$ And I have no idea how to get there. $$Zorich 3.2$$ $$Problem 9.a$$
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how $\sqrt x \sqrt{1+\epsilon} \approx \sqrt x \left( 1+\frac{\epsilon}{2} +O(\epsilon ^2) \right)$

I am looking for some explanation of how the following was completed. $$\sqrt{x(1+\epsilon)}= \sqrt x \sqrt{1+\epsilon}\approx \sqrt x \left( 1+\frac{\epsilon}{2} +O(\epsilon ^2) \right)$$ I ...
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$f(x) = \frac{1}{x^2} \quad x_0 =3$ find Taylor expansion using term-by term approach

I have a function and I have to find the taylor series expansion using term by term integration and differentiation: $$f(x) = \frac{1}{x^2} \quad x_0 =3$$ Well, my concern is that I only know the ...
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Points of Confusion About Second-Order Taylor Formula of Taylor's Theorem For Many Variables

My textbook has written the following for the second-order Taylor formula of Taylor's theorem for many variables: $$f(\mathbf{x}_0 + \mathbf{h}) = f(\mathbf{x_0}) + \sum_{i = 1}^n h_i \dfrac{\...
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1answer
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Taylor's Theorem Proof: Confused About the Validity of Two Inequalities

My textbook gives the following proof of the single-variable version of Taylor's theorem: As promised, we begin with the Fundamental Theorem of Calculus, written in the form $$f(x_0 + h) = f(...
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4answers
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Taylor series expansion of $f(x) = \frac{1}{x^2}$

Expand $f$ in a Taylor’s series about a center $b = 1$, so that $f(x)=\sum_{n=0}^\infty a_n(x-1)^n $. I know a Taylor series takes the form $$\sum_{n=0}^\infty \frac{f^{n}(a)}{n!}(x-a)^n$$ but I don't ...
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Integral Expression Without $dx$, $dy$, $d\tau$, etc.

My textbook gives the following proof of the single-variable version of Taylor's theorem: As promised, we begin with the Fundamental Theorem of Calculus, written in the form $$f(x_0 + h) = f(...
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Sum of $\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - …$

I have to find the sum of : $$\frac{x^2}{2*1} - \frac{x^3}{3*2} + \frac{x^4}{4*3} - \frac{x^5}{5*4} +\cdots$$ So far I have : $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \, x^{n+1}}{(n+1)(n)}$$ which ...
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1answer
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Taylor expansion of $\log \frac{1+\exp(u+h)}{1+ \exp(u)}$

Hjort and Pollard write in their article Asymptotics for minimisers of convex processes that the following expansion holds for all $u$ and $u+h$, in terms of $\pi(u) = \exp(u)/\{1+\exp(u)\}$: $$\...
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Finding the Value of M for a Taylor Inequation

The initial function I have is around a = π/2. Which then can be written as . The issue I have is when trying to find the minimal degree of n such than |f(π) - T(π)| <= 1/100, I want to apply ...
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Verify this Proof: $\sum_{k=0}^{n}{n\choose k}=2^n$ via Taylor series

I've found this proof which I am quite proud of. Am I missing anything? Theorem: $$\sum_{k=0}^{n}{n\choose k}=2^n$$ Proof: Let $\beta\in\{t\in\Bbb R:t>0\}$, and $f(x)=x^\beta$ be continuous on ...
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1answer
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Conjecture: $(x^\alpha+p)^\beta=\,_1F_0(\beta;;x^\alpha+p-1)$ and ideas for proof

Conjecture: For $|z|<1,\,\alpha,\beta,p\in\Bbb R$ $$(z^\alpha+p)^\beta=\,_1F_0(\beta;;z^\alpha+p-1)$$ I found this formula by noting that $$z^\alpha=\,_1F_0(\alpha;;z-1)$$ Via the simplification of ...
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1answer
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Approximate Trig Functions without the use of Taylor Series

I am familiar with how a trig function, i.e. $\sin(x)$, can be approximated by a MacLauren series; \begin{align} \sin(x_0) &\approx \sin(0) + \cos(0) x_0 - \frac{1}{2}\sin(0) x_0^2 - \frac{1}{3!}\...
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How can I relate this question to Series expansion of functions? [closed]

if $\,x^3+y^3+xy-1=0\,$ then show that $y=1-\frac{x}{3}-\frac{26}{81}x^3+....$
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1answer
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Is the similarity between tanh and normal distribution just coincidence?

So, explaining to someone why tanh is used in machine learning (i.e. it squashes an open range to -1..+1, and changes most rapidly around 0), I brought up $\frac d{dx}$ $tanh(x)$, and it looks just ...
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4answers
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Proving approximation of $\text{erf}$ with Taylor expansion

I am asked to show that $$\text{erf}(x) \approx 1 - \frac{1}{\sqrt{\pi}}\frac{1}{x}e^{-x^2}$$ in a computational project. Numerically it is really easy to show that this approximation makes sense. ...
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2answers
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Textbook Taylor's Theorem Proof: Integration by Parts Notation

My textbook begins its proof of Taylor's theorem as follows: $$f(x_0 + h) = f(x_0) + \int_{x_0}^{x_0 + h} f'(\tau) \ d \tau$$ Next, we write $d \tau = -d(x_0 + h - \tau)$ and integrate by ...
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Modified Bessel function of order 0 and 1 question

@skbmoore Prove that $$\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}\,r^n}{(n!)^2}=\frac{\sqrt{\pi }}{2} e^{r/2} \left((r+1) I_0\left(\frac{r}{2}\right)+r I_1\left(\frac{r}{2}\right)\right)$$ ...
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Using Taylor series expansion of random vectors to find expectation

Assume that the second-order Taylor expansion about point $x_0$ is given by $$g(x) = g(x_0) + g'(x_0)(x-x_0) + \frac{g''(x_0)(x-x_0)^2}2 + \text{remainder}$$ Let $\bf X$ be a $k\times1$ vector with ...
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What exactly does it mean to say “$dv=4\pi r^2\,dr$ can be thought of as the spherical volume element between $r$ and $r+dr$”?

In my textbooks and lectures (I'm a second year physics student) I often come across statements such as $$``dV=4\pi r^2dr\text{ can be thought of as the spherical volume element between }r\text{ and }...
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Developing a fraction of infinite sums into a power series for integration

I want to express the function $$f(x) = \frac{1+a_1 x + a_2 x^2 + a_3 x^3 + \dots}{b_1 x + b_2 x^2 + b_3 x^3 + \dots + \log(x)\left[ c_2 x^2 + c_3 x^3 + c_4 x^4 + \dots \right]} $$ as a (generalized) ...
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1answer
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Inversion of Taylor series

A proof I'm reading states without further explanation that $$ -\frac{1}{2}\delta^2 = \frac{1}{2} \zeta^2 + \frac{\alpha}{6} \zeta^3 + \frac{\beta}{24} \zeta^4 +\cdots $$ can be inverted into $$ \...
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1answer
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Taylor Series with Big-O Notation

Many times in lectures, I have seen the use of $\mathcal{O}$, especially when writing out Taylor series and so I would like to clarify if my understanding on its use is correct. I came across this ...
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Tricks for estimating $ \lim_{x\rightarrow 0} \frac{d}{dx} \bigl(-\frac{1}{x} \ln\bigl(1 + \frac{(e^{-xu}-1) (e^{-xv}-1)}{e^{-x}-1} \bigr) \bigr)$

I'm trying to find a Taylor approximation of $ f(x) =-\frac{1}{x} \ln\left(1 + \frac{(e^{-xu}-1) (e^{-xv}-1)}{e^{-x}-1} \right) $ at $x = 0$. For the derivation part Wolfram returns a quite a ...
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How to get Taylor Series of $\sin \frac{x}{1-x}$

I know that $\displaystyle\sin x = \sum_{k=0}^{\infty}\frac{\left( -1 \right) ^kx^{2k+1}}{(2k+1)!}$ But how to get transformation to get Series about x?
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High Order Derivative

Find the ninth derivative of the following function at $x=0$: $$f(x) =\frac{\cos\left(4x^4\right)-1}{x^7}$$ So I did all the manipulations and I got the following Maclaurin Series: $$\sum _{n=0}^{\...
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2answers
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Identity $f(x)=f(x^{*})+\int_{0}^{1}{∇f(x+te)^{T}e\space dt}$

This is a theorem from the book Iterative Methods for Optimization. Theorem 1.2.1. Let f be twice continuously differentiable in a neighborhood of a line segment between points $x^{∗}, x = x^{∗} + e ∈ ...
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1answer
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Use Lagrange Remainder to estimate the error term

The question asked us to find an upper bound for the absolute error in estimating $\sqrt{x}$ for $3\leq x \leq 5$ with the quadratic $0.75+0.375x-0.015625x^2$. A close look reveals tha the quadratic ...
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Calculating Taylor Polynomial and its error

there is an extension question I am trying to solve for my exam revision. It reads Use an appropriate Taylor polynomial for sin x and apply the Taylor’s formula for the remainder to approximate $\sin(...
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Proof Check: The Product rule via Taylor Series

I've been working on the following proof of the product rule for derivatives using Taylor series. Are there any holes? Thanks. Setup: Let $I\subseteq \Bbb R$ be some interval. Let $f,g:\Bbb R\mapsto\...
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1answer
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Does continuity of $f(x)$ imply continuity of $f^{(n)}(x)$? And other questions.

I'm working on a proof of the product rule for derivatives using Taylor Series, and the following questions have come up in the process. I'm suspecting that the answer to each of these questions is ...
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1answer
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Prove that Taylor Series of a polynomial is the polynomial

Recently I've asked a question related with the Taylor Series of the Lagrange Basis functions. In the comments someone wrote that the Taylor series of a polynomial is the polynomial. I know that ...
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f(x)=arctan⁡(x), c=0, [-3,3]

Calculate the Taylor polynomials $P_0 , P_1 , P_2 , P_3$ , and $P_4$ for the given function centered at the given value of $c$. Then graph the function and the Taylor polynomials on the given interval....
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How can the Taylor expansion of sine converge for all x?

So, I am writing a C++ program that sum the series expansion of sin(x). But for large values of x my program fails. $\sin(x)=x-(x^3/3!)+(x^5/5!)-...=$$\sum_{n=0}^n (-1)^n\frac{x^{2n+1}}{(2n+1)!}$ ...
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What is the expansion of $\log(N+x) = \log(N) + [\dots\text{blank}\dots] $? ($N \in \mathbb{R}+$ and $0 \leq x \leq 1)$.

I'm working on a math problem which might be solvable if I can re-express $\log(N+x)$ as $\log(N) +$ 'something. The problem I am having with the Taylor series expansion about $x=0$ is that it ...
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1answer
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Taylor series for $f(z)=\frac{\exp(z^2)}{z^3-1}$ with recursive coefficients

Let $f(z)=\frac{\exp(z^2)}{z^3-1}$, $z\in\mathbb{C}$, and show that the Taylor series for $f$ around $0$ has the form $$ f(z)=\sum_{n=0}^\infty a_k z^k$$ where $a_0=-1$, $a_1=0$, $a_2=-1$, and $a_{...
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Taylor polynomial for $f(x, y) = \arctan(\frac{y}{x})$

Given the function $f(x, y) = \arctan(\frac{y}{x})$ defined for $(x,y)$ with $x>0$, how can we find Taylor polynomial in point $x_0=0$? As I understood, Taylor polynomials for $n=1$ for functions ...
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Convergence radius estimator with three consecutive terms

The well known ratio estimator for the convergence radius of a Taylor series is $\left| \frac{c_k}{c_{k+1}}\right|$ and it uses two consecutive terms in the Taylor series, but does not always work. E....
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2answers
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Finite Taylor series of Lagrange Basis functions

Lagrange basis functions are defined as follows $$L_{j}(x) = \prod_{i\neq j} \frac{x-x_{i}}{x_{j}-x_{i}} $$ then $$\ln\Big(L_{j}(x)\Big) = \ln\Big(\prod_{i\neq j} \frac{x-x_{i}}{x_{j}-x_{i}} \Big) = ...
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1answer
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Power Series / Taylor Expansion of $\frac{1}{\left(1-t^2\right)^{\frac{1}{2}}}$

The question: $\left(\frac{1+t}{1-t}\right)^{\frac{1}{2}}\:=\:\sum _{n=0}^{\infty }\:a_nt^n$ The question asks for what is $a_n$ These are the steps I've done: $\left(\frac{1+t}{1-t}\right)^{\frac{...
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1answer
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Taylor expansion of $x^2(t)$?

This function for air resistance (drag) supposedly shows a Taylor expansion. I understand the basics of Taylor expansion, but can't see how I would get this answer. If someone could elaborate and ...
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1answer
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Unbiased Estimation of Sum of Reciprocals over a Symmetric Distribution by Taylor Expansion

Random variable $X$ follows a symmetric and unkown distribution. $\lbrace x_n \rbrace$ are a large (~$10^6$) sample drawn from $X$ Expectation $a = E[X]$ is known. Consider the taylor expansion of $f(...
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1answer
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$h$ in Textbook Definition of Taylor's theorem for One Variable: $f(x_0 + h) = \dots$

When discussing Taylor's theorem for one variable, my textbook says the following: For a smooth function $f : \mathbb{R} \to \mathbb{R}$ of one variable, Taylor's theorem asserts that $$f(x_0 ...