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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Confusion related to Taylor series approximation

I found this Taylor series approximation given by $f(x_{\alpha}) = f(x) + \nabla f(x)'(x_{\alpha}-x) + o(||x_{\alpha}-x||)$. I didn't get how this $o(||x_{\alpha}-x||)$ term came from. Can anyone ...
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1answer
161 views

Explaining and using the $N$-term Taylor series for $\sin x$

So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer: Explain why the Taylor series containing $N$ ...
4
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1answer
150 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
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1answer
245 views

What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
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3answers
409 views

Taylor expansion and error?

This came up in a part of the proof. $-\log(1-x)$ is $x$ and then want to calculate the error of this. The idea is that taylor series of $-\log(1-x)=x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+...$ We have ...
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2answers
44 views

Taylor evaluation in a product solving a limit

I have the following function, which I am supposed to evaluate: $\lim_{x \to 0}{\frac{(e^{-x^2}-1)\sin x}{x \ln (1+x^2)}}$ My though is to replace sin x by its Maclaurin polynomial, as such: ...
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2answers
1k views

Natural Logarithm Taylor Series Expansion

f(x)=x$^3$ln(1+2x) Write the first four non-zero terms of the Taylor Series for the above function with x centered at a=0. Using this model: ln(1+x) = Σ$\frac{(-1)^{k}(x)^{k}}{k}$ I get the ...
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1answer
276 views

How many iterations of Taylor series for n correct decimal digits

I'm using Taylor series to estimate trigonometric functions. So I need to know exactly how many iterations of Taylor series (say for sine) are needed for n decimal digits precision? (I'm writing a ...
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1answer
189 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
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1answer
55 views

Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
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1answer
56 views

Does the Taylor Polynomial approximation work for non-convergent functions?

The approximation of $f(x)$ by $P_{n}(x)$ at $c$ has an error of $R_{n}(x) = \frac{f^{n+1}(z) (x-c)^{n+1}}{(n+1)!}$. Does this work for any $(n+1)$ differentiable function even if it doesn't have a ...
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2answers
511 views

Finding Taylor's expansion for $f(x) = \sqrt{1 + x} -\sqrt{ 1 - x}$

I know I have to find the derivatives of $ f(x) $ (i.e. $f'(x)$ ..) but I'm confused about what to do afterwards .
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2answers
205 views

Is there a closed form expression for the Taylor series of exp((f(z))?

Given a holomorphic function $f(z) = \sum_{k=0}^\infty f_k z^k/k!$, is there a readable formula for the Taylor series of $\exp(f(z))$? Using the chain and product rules, one can obtain $$\partial_z ...
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1answer
114 views

Nonlinear initial-boundary value problems using Taylor expansion of parameter

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) ...
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1answer
236 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
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0answers
93 views

A function with an identically zero Maclaurin series

We just recently went over Taylor's theorem in my analysis class, and my professor gave the function $$ f(x) = \begin{cases} e^{-1/x^2}, & \text{if }x \not= 0, \\ 0, & \text{if }x\text{ = ...
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1answer
65 views

Rotated functions and taylor series

If one rotates a function such as the sine function about the origin, is there a general method to find the taylor series for the rotated function? Assuming of course that the rotated function is ...
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1answer
235 views

Algorithm for estimating $\beta$ using a Taylor series expansion

I am working on the following question for a mathematical economics class. Consider an econometric model: $$y_t=f(x_t,\beta) + e_t,t=1,...,T$$ where $\{ e_t \}$ is a sequence of mean-zero ...
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2answers
254 views

Find the two variable Maclaurin series for $f(x,y) = e^{x+y}$

I'm shaky with Taylor/Maclaurin series, and I've been over and over my book and notes and still feel like I'm at square one...
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0answers
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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 $a=5$ ...
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1answer
75 views

Proofs using Taylor Series Expansion

I would just like some help on the theory of maths. If a question asks for proof of a function using the Taylor Series Expansion, can you use the Maclaurin Series? Is using the Maclaurin Series ...
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2answers
2k views

Taylor Series Expansion for $\sin^2(\omega t)$

What are the first few terms for the Taylor Series Expansion for $\sin^2(\omega t)$? $(\omega$=$2\pi f$) If you could show some working, that would be helpful
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91 views

show that $\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi x)=-(2\pi)^{2k-1}2^{2k}(2^{2k}-1) \frac{\left | B_{2k} \right |}{2k} $

I try to prove the relation between Polygamma function and Bernoulli numbers but I faced this problem,is how to show that $$\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi ...
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2answers
53 views

Taylor series problem

I have this equation: 960 - 84.60 * ((1-(1+i)^-12)/i) == 0 I simplify ( 1+i)^-12 with a Taylor series ...
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1answer
282 views

Why do we use big Oh in taylor series?

In the taylor series for sin(x), we write: $$ \sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120} + O(x^7) $$ Meaning that $\sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120}$ and terms of order $x^7$ and ...
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4answers
181 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
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2answers
88 views

What does the taylor series give us ?

To be more clear if we use the taylor series for x=2 it will give us an approximation of f(2) ? And why do we stop "adding" the f'''(a)/3! ...? Is there a rule that tells us when to stop ?
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1answer
42 views

A basic doubt on derivatives

I have one question regarding differentiation : 1) Why in the definition of Taylor's series it requires the function to be "continuously" differentiable $m$ times in $[a,b]$? The book I am following ...
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3answers
890 views

Derivation of multivariable Taylor series

I am having trouble grokking why it is, assuming that the function is analytic everywhere (and many other assumptions that I am, no doubt, naively assuming), that this is true: ...
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1answer
42 views

Taylor Series and equation

I have this equation: $$960 - 84.60 \cdot \frac{1-(1+i)^{-12}}{i} = 0$$ I simplify $( 1+i)^{-12}$ with a Taylor series $( 1 + x)^a$. but I obtain $i = 0.087201167$ but the real result should be $i ...
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1answer
54 views

Taylor expansion of $((H+\epsilon A)^T R^{-1} (H+\epsilon A))^{-1} (H+\epsilon A)^T R^{-1}$

I have seen a kind of contradiction in a paper and I decided to rewrite the equations... Could you please help me to be sure about what I am doing... Let us define $H^\dagger \triangleq ...
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3answers
405 views

Complex Analysis Taylor Series

So the problem states: "Say f(z) := log z is the principal branch of the logarithm (the primitive of 1/z on the region C(-infinity,0]). Show that the Taylor series of f(z) about $z_0 = -1 + i$ takes ...
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2answers
98 views

First order multivariate approximation

To demonstrate that $\nabla\!_{\hat{\boldsymbol u}}\,f(\boldsymbol{x}) \equiv \left \langle \hat{\boldsymbol u}, \nabla f(\boldsymbol{x}) \right \rangle$ I plug a first order expansion of ...
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1answer
196 views

How to compute the Lagrange remainder of a Taylor expansion

In a Taylor expansion with Lagrange remainder, how can I compute the remainder $R_n(x)$? How to find the $(n+1)$th derivative? Please explain it with elementary functions like $\ln(x),\sin x$ and ...
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1answer
58 views

Talor series of fraction addition of common function

What is the typical trick for finding the taylor series of a common function that is in the denominator when adding a constant. eg: $$f(x)=\frac{1}{e^x-c}$$ I know you can write ...
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4answers
678 views

The Idea Behind Taylor Series

I understand that they are viewed as approximations, but was that Taylor's original hope? Assuming that a function can be written as a power series seems to me to be a wild assumption, without some ...
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0answers
161 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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0answers
70 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 ...
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0answers
47 views

What is the Taylor expansion of $W( \vec{n}+ \varepsilon \vec{\eta}, \nabla \vec{n} + \varepsilon \nabla \vec{\eta})$

What would the Taylor expansion of $$W( \vec{n}+ \varepsilon \vec{\eta}, \nabla \vec{n} + \varepsilon \nabla \vec{\eta})$$ be about $\vec{n}$, to order $\varepsilon$? Where $$\nabla ...
4
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2answers
215 views

Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
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1answer
110 views

Use Taylor sequence write approximate value

Use Taylor sequence write approximate value: $$ \sqrt{9.5} $$ Estimate the error approximations three components. Which function should I expand?
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87 views

Prove that $\sin(x^2) = \mathcal{o}(x)$

I tried to do this like that: $$ \sin(x^2) = \mathcal{o}(x) \iff \lim_{x \to 0} \frac{\sin(x^2)}{x} = 0$$ we could get $\sin(x^2)$ from Taylor series. For $x_0 = 0$, $T_n = 0$ for every $n$. So from ...
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4answers
227 views

Trigonometric Coincidence

I Know that using Taylor Series, the formula of $\sin x$ is $$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$ and the unit of $x$ is radian (where $\pi/2$ is right angle). However, the ratio of the circumference ...
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2answers
55 views

Function and Maclaurin series

Function $f(x)=\frac{x^2+3\cdot\ e^x}{e^{2x}}$ need to be developed in Maclaurin series. I can't find any rule to sum all fractions I've got...so any suggestion that helps? Thanks
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1answer
203 views

How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
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2answers
92 views

How can we determine the number of terms which we have to take in a series to get a particular accurate?

As I remember , two days ago , there was a question ( here ) asks for calculating this limit $\displaystyle \lim \limits_{x\rightarrow \infty } \frac{x^3}{e^x}$ and the question was answered . of ...
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1answer
451 views

Why is$ (1+\frac{1}{n})^n=e$ when n goes to infinity? [duplicate]

Why is $\lim\limits_{n\to\infty}(1+\frac1n)^n=e$? I think it involves $\sum\limits_{n=0}^\infty\frac1{k!}=e$ but not sure how to get from one to the other.
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2answers
106 views

Taylor expansion

Is there an easier way to do a Taylor expansion of $e^{u^2+u}$ than do derivatives or substitute and then use Newton's binomial? For example, expanding until the $4$th term: $$e^{u^2+u}=1+u^2+u+ ...
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1answer
181 views

Thinking through a Taylor error bound for arcsine

In lecture, we went through solving a Taylor error bound for arcsine. I followed most of it except for where it talks about the odds divided by the evens divided by $2n+1$ gaining in accuracy by a ...
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1answer
93 views

Calculating $\arctan(3)$ using Taylor series

I'm trying to get a Taylor series equivalent for $\arctan(3)$, but the standard definition for $\arctan(x)$ is restricted to $|x| \le 1$. How can I get a Taylor series for this expression?