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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Convergence radius argument

I'm studying complex function theory and I ran into this argument made by my prof but I can't really wrap my head around it. Set $f(z):=\frac{1}{7+z^2}$ Now notice $\sum_{n=0}^\infty ...
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1answer
59 views

Taylor series at two different points

If I have a function $y(x)$ for which there is a Taylor series about $x=1$ that has an infinite radius of convergence, and I also have a Taylor series for $y(x)$ about $x=0$ (with unknown radius of ...
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18 views

Function property of $o$

I have a doubt regarding $o$-function. Could we write $o(\|\theta h)\|)=\theta \ o(\|h\|)$ ?
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35 views

calculate limit with Taylor and L'hopital

i need to calculate a limit, and our teacher told us to use L'hopital and Taylor aproximations. $$\lim_{x\to 0}\left({\sin(x)\over x}\right)^{1/x^2}$$ and that must be equal to; $$e^{1/6}$$ and i ...
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24 views

Binomial random walk: Taylor series

Consider the following probabilities associated with a binomial random walk: $$ p(y',t') = \dfrac{1}{2}p(y' + \delta y, t' - \delta t) + \dfrac{1}{2}p(y' - \delta y, t' - \delta t) $$ What's ...
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34 views

Estimating $\int_0^1\sin(x^2)dx$ using the Taylor expansion of $\sin(x)$

Problem: a) Find the Taylor polynomial $T_6(x)$ for $f(x) = \sin(x)$ about $x=0$. I found this to be $x-\frac{x^3}{6} + \frac{x^5}{120} + O(x^6)$. b) Use this to estimate $\int_0^1\sin(x^2)dx$ with ...
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1answer
148 views

taylor expansion in cylindrical coordinates

If I have a function Y($r$,$\theta$) in cylindrical polar coordinate system, then how do I Taylor expand this function around some point ($r_0$,$\theta_0$)? I want the exact formula for Taylor ...
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60 views

Taylor series applied to logarithm

Following formula at $k=0$ yields $\frac{1}{2i}\log\frac{1+i}{1-i}=\frac{\pi}{4}$. $$\sum_{n=0}^\infty\frac{i^{2n}}{2(n+2k)+1}$$ At $k\in\Bbb N$, Mathematica throws out $\frac{1}{2} ...
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4answers
58 views

Taylor Series $(x+2)/(2-3x)$ at $x=2$ [closed]

How can I find Taylor series for $$\frac{(x+2)}{(2-3x)}$$ at $x=2$?
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1answer
30 views

Taylor series $\ln(x+3)$ at $x=1$

Taylor series $\ln(x+3)$ at $x=1$ I am a little confused if both ways are correct: $y=x-1$ $$\ln(y+4)=\ln(4) + \ln (1+y/4)=...=\ln(4)+\sum_{n=1}^\infty(-1)^{n-1}(1/4)^n\frac{(x-1)^n}{n}$$ or ...
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1answer
28 views

An Example of Taylor Series

Consider a positive function $f(x)$ and suppose that we would like to approximate its value around some point $x_0$. One way to do so is to use two-term Taylor series expansion as follows. $$ f(x) ...
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2answers
80 views

Higher Order Terms in Stirling's Approximation

Some websites and books give stirling approximation as $$n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \left( 1 + O \left(\frac{1}{n} \right)\right)$$ However when I check their derivations most ...
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1answer
24 views

Taylor expansion of $\frac{(-1)^n}{\ln n(1+\frac{1}{n\ln n}+o(\frac{1}{n\ln n})}$

I can't get the right terms: $$\frac{(-1)^n}{\ln n + \frac{(-1)^n}n + o(\frac1n)}=\frac{(-1)^n}{\ln n} - \frac1{n\ln^2 n}+o\left(\frac1{n\ln^2 n}\right)$$ My thoughts $$\frac{(-1)^n}{\ln ...
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1answer
33 views

How can get the series of $\log(x/(x-1))$ at $x=\infty$

When I used the Wolfram to give me the Taylor series of $\log(x/(x-1))$, I was amazed of the result. The Wolfram give me a Laurent series at $x=\infty$ as follow ...
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1answer
43 views

What is the remainder of $|e-\sum_{j=0}^n{1\over j!}|$?

I have to find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$, but I want to do it with the remainder. I know that it is ${e^c\over (n+1)!}$ where $0<c<1$, but how do you get ...
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1answer
38 views

Where does the $\mathcal{O}$-term come from in Taylor series?

I understand Taylor series in general, but I've always been a bit uncertain about the $\mathcal{O}$-term, when I see it used in Taylor series. For example in one of my study materials I have the ...
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245 views

Taylor series for $\sqrt{x}$?

I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particular reason no one shows Taylor series for exactly ...
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1answer
41 views

Find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$.

Find the smallest $n$ such that $|e-\sum_{j=0}^n{1\over j!}|<0.001$. I know it has something to do with remainder, or Taylor expansion, but I am week in this material. What I did is merely: ...
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1answer
156 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
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2answers
47 views

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$.

Find $\lim_\limits{n\to\infty}\sin(\pi\sqrt[3]{n^3+1})$. I am trying to find it using Taylor series. What I did so far is: $\sqrt[3]{z+1}=1+O(z)$ (I really can't tell when I should be done ...
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3answers
33 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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40 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ .

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
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2answers
34 views

MacLaurin Series with Variable in Denominator

A friend of mine was talking about how finding MacLaurin series for functions with variables in the denominator might prove difficult without tables. We started making lots of crazy problems, but one ...
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2answers
31 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
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0answers
165 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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4answers
72 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
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0answers
19 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
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1answer
41 views

Functionals' Taylors Theorem

Consider functional $F:B\to \mathbb{R}$, where B is a Banach space eg. $B=H^{1}(\mathbb{R}^{d},\mathbb{C})$. Then Taylor's theorem for functionals is: Suppose that the line segment between u ∈ ...
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1answer
27 views

Number of derivatives in a taylor series expansion

I would like to confirm if the number of derivatives we need to calculate in a specific order of a taylor series expansion is the sum of the multinomial coefficient of that order: $$ f:\mathbb{R}^k ...
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1answer
62 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
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1answer
32 views

Taylor theorem for f(x+h)

I am following a proof that applies Taylor's theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf) I am not understanding one of the terms explained on ...
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14 views

A simple question about Delta Method's demonstration.

Suppose that $\sqrt{n}(X_n-\mu)\stackrel{D}{\longrightarrow}X$ and consider $g:\mathbb{R}\rightarrow\mathbb{R}$ a function such that first derivative $g'$ is continuous in a neighbourhood of $\mu$, ...
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4answers
106 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
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3answers
53 views

taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
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1answer
31 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...
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2answers
57 views

What is the coefficient of $(z-\pi)^2$ in Taylor series expansion of $\sin (z)/ (z-\pi)$

I want to determine the coefficient of $(z- \pi)^2$ in Taylor series expansion of $f(z)=\sin (z)/ (z-\pi)$ if $z \neq \pi $, $-1$ if $z=\pi$ around $\pi$. How can this be done? I don't know how to do ...
2
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1answer
53 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
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1answer
41 views

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$

Find the coefficient of $x^{4}$ from $(1+x)^{1/3}$ Should I use the formula $C(n,k) = n!/[k!(n-k)!]$? And what is the solution of this problem?
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24 views

taylor series approximation of e function

in the equation $$e^{y(x)}=1+2x-\frac{y(x)}{1-x}$$ $y(0)=0$ because using the taylor series and by comparing the coefficients we obtain $$1+y(0)=1-y(0)$$But why is using the taylor series allowed. ...
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1answer
54 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
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1answer
29 views

Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$

Maclaurin Expansion of $\ln(1+4x^2+4x)$ in terms of $\sum a_k x^k$ The question has written to $x^2$ term before the $x$ - does that have anything to do with how to solve the problem? Am I meant to ...
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1answer
42 views

Extending Taylor's theorem from one to several variables

In my calculus class we are dealing with Taylor´s theorem in several variables. When we were looking at the function $f(x,y)=\sin(xy)$ my teacher said that instead of applying the theorem in several ...
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1answer
53 views

Using $\ln (\cos x)=\frac{-x^2}{2}-\frac{x^4}{12}+…$, approximate $\ln 2$ in terms of $\pi$

Using $f(x)=\ln (\cos x)=\dfrac{-x^2}{2}-\dfrac{x^4}{12}+\dots $, approximate $\ln 2$ in terms of $\pi$. I know $\cos(x)$ will never be two - so what can I actually substitute in to get something ...
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1answer
36 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
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1answer
30 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
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1answer
26 views

Functional expansion

I am confused by this expansion in Landau and Lifshitz: First, they define $\textbf{v}' = \textbf{v} + \textbf{$\epsilon$}$. So for a function $L$, $$L(v'^2) = L(v^2 + ...
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1answer
136 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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2answers
34 views

Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
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0answers
16 views

How can I show the remainder of this Taylor polynomial $R(h)/h^2$ goes to $0$ as $h$ goes to $0$?

Given the function $f(x, y) = 1/(2 - x - y^2)$ I found that the second-degree Taylor polynomial is $P(x, y) = 1/2 + x/4 + x^2/4 + y^2/2$ How can I show the remainder $R(x, y) = f(x, y) - P(x, y)$ ...
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1answer
39 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...