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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Taylor series of $\sqrt{\ln\left(\frac{1}{x}\right)}$

I am trying to compute the Taylor series of: $\sqrt{\ln\left(\frac{1}{x}\right)}$ I have computed the derivatives and evaluated them in $x=1/e$ but I cannot find the formula for the sequence of the ...
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3answers
71 views

Taylor expansion at $x=0$ of $\ln(1/(1-x))$

Hello I am having some trouble with the taylor expansion of $$f(x)= \ln \frac1{1-x}$$ Would it be correct to treat the inner part as the following geometric series? ...
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5answers
102 views

Taylor series for $\sinh1$

I am doing taylor series and I want to do it on $\sinh1$. is there a way to make this problem really simple before I begin? note: $\sinh x= \cfrac{e^x - e^{-x}}2$ Any ideas are really helpful ...
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2answers
52 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
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0answers
9 views

approximate a function by linear combination of its asympototics with fractional argument

Suppose that $0\le a \le 1$ and $0<f(x), g(ax)<\infty$ for $x\ge 0$ and $f(x)\to g(x)$ when $x\to \infty$. And also $g(a x)<g(b x)$ if $a<b$. Question: Can we use $g(a_k x)$ $(0\le a_1\lt ...
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1answer
29 views

Approximating a curve with a parabole at a given point

I wonder if such a task is possible: we have a curve defined implicitly with: $x^4+y^3-xy-1=0\qquad(1)$ I want to find a parabola in a general form of: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ ... which ...
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2answers
73 views

3rd Order Taylor expansion of $e^x\cos(y)\sin(z)$

I'm looking for the 3rd.-order Taylor approximation of $(x,y,z) \mapsto e^x\cos(y)\sin(z)$ at $(x_0,y_0,z_0) = (0,0,0)$ I've got this piece of advice at hand: $\quad\textit{Use the Taylor series ...
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1answer
26 views

Reference Request: Vector valued Taylor formula

It is well known that for analytic function defined on interval $I$ we have $$ f(x)=\sum^{\infty}_{k=0}f^{k}(0)\frac{x^{k}}{k!} $$ and for function defined on $I^{n}\rightarrow \mathbb{R}$ we have $$ ...
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1answer
19 views

Linear functional vs. map

A few days ago we were briefly discussing Taylor's theorem in higher dimensions in the lecture. Referring to the expression $f(x)=f(a)+Df(a)(x-a)+$higher order the lecturer said that in general $Df$ ...
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2answers
33 views

Evaluating a taylor series around a given point

So I'm having some trouble with the problem: Given that $\ln(x+1)=\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}x^{n}, -1<x\leq 1$, find the Taylor series of ln(x) around 3. For which x is this series ...
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1answer
77 views

Quick question on convergence

Shouldn't the radius of convergence be defined as: $$\frac{1}{R} = \lim_{n\rightarrow \infty} \left(a_n \right)^{\frac{1}{n}}$$ Not sure what they are doing above..
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0answers
58 views

Taylor series approximation using for a pdf

I have an question which links Taylor series to expectation and variance, but I'm really not sure what it's asking me to do. X is exponential with rate 1. The question asks me to use a three term ...
4
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2answers
73 views

Taylor Series Representation of $e^{1-\cos(x)}$

Hello I was wondering how to simplify this Taylor Series $$ e^{1-\cos(x)} =\sum_{k=0}^\infty\frac{(1-\cos(x))}{k!} ^k\ $$ to where I can write out the first couple of terms which are $ ...
3
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1answer
65 views

gradient of norm square of a random vector

Let $g(w)= \|Y_n - f(w,X_n) \|^2$ where $f:\Bbb R^d \times \Bbb R^m \to \Bbb R^k : w \in \Bbb R^d$. What is the gradient of $g$ ? $X_n$ and $Y_n$ are random vectors. Basically, I want to find ...
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1answer
18 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
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1answer
60 views

$f(x)=e^x$ be approximated by Taylor's polynomial of degree $n$ at the point …

I am stuck on the following problem: Let $f(x)=e^x$ be approximated by Taylor's polynomial of degree $n$ at the point $x=\frac12$ and on the entire interval $[0,1]$. If the absolute error in this ...
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1answer
48 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
3
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2answers
97 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
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199 views

Limit of $\dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ as $x \rightarrow 0$

Find $\lim_{x \to 0} \dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ I came across this limit a long time ago and could easily obtain a straightforward solution ...
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1answer
55 views

Show that $|f'(x)| \le \frac{2M_0}{h} +\frac{hM_2}{2}$ and $M_1 \le 2\sqrt{M_0M_2}$

Let $f$ be a twice derivable function and $M_i =\sup_{x \in \mathbb{R}} |f^{(i)}(x)|$ and $|M_0|, |M_2|<\infty $. Preferably using the Taylor series on the interval $[x,x+h]$ show the following ...
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0answers
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Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
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1answer
27 views

Finding an upper bound error of a Maclaurin polynomial.

Using a 3rd order Maclaurin polynomial, find an upper bound on the error when log(1+x) is approximated by a 3rd order polynomial for |x|<= 0.1. For some reason I keep getting a different answer ...
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1answer
34 views

Prove the taylor series of $ \cos(2z)$

First i turned $$\cos(2z) = \frac{e^{2iz} + e^{-2iz}}{2}$$, then using the taylor series of $$e^{z}$$I calculated the taylor series of both arguments. $$\frac{e^{2iz}}{2} = \sum_{n=0}^{\infty ...
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1answer
66 views

Question about infinitely many times differentiable function.

Could you please give me some hint how to solve this problem: Suppose $f(x)$ is infinitely many times differentiable function on R, $f(0)=f'(0)=f''(0)=0$. Prove : for all $A>0$ exists some ...
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0answers
18 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 ...
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4answers
70 views

Taylor Series - general 'what' and 'why' questions

I am a little confused with the Taylor Series at the moment, so please forgive me for my very basic questions. If we were to approximate a function, say $cos(x)$, I let $f(x)=cos(x)$ And I have been ...
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1answer
59 views

Why is the series expansion of $\sin(x)$ at $x=n\pi$ different to the expansion of $\sin(2x)$ at $x=\frac{n\pi}{2}?$

I'm trying to do a series expansion of $\sin(2x)$ about the point $x=\frac{n\pi}2$ where is an integer. I thought that the expansion would be the same as for $\sin(x)$ about the point $x=n\pi$ but ...
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0answers
66 views

Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
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1answer
46 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
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0answers
27 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
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3answers
33 views

Find the first three terms of the taylor expansion of $\frac{cos(z)}{1 + z^2}$

The question is: Find the first three non zero terms for the taylor series for $\frac{\cos(z)}{1 + z^2} $ around $z_0 = 0$ What I've done so far is let $f(z) = \frac{\cos(z)}{1 + z^2}$ Then I let ...
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2answers
28 views

Complex functions and Taylor series

Find the Taylor series arround $z_0=0$ write radius of convergence a) $f(z)=\cosh(z)$ b) $f(z)=\log(z+1)$ I don't know how it works with the complex functions. Could you show me the workflow? I ...
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2answers
79 views

Is $\cos(\frac{\pi}{3})$ exactly equal to 0.5 or approximately equal to 0.5

We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$, but the expansion for $\cos(x)$ is as follows: $$ \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ So this would make $$\begin{align} ...
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3answers
32 views

Proving an inequality using Taylor's Theorem

I need to show that $ x^{1/3} < \frac{1}{3}x + \frac{2}{3} \forall x \in (0,1)$. I have been given the hint to consider the expression $\frac{1}{3}x - x^{1/3}$, but the Taylor Series centred at ...
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1answer
55 views

How do I compute the Taylor Series for $\arctan(x)$?

I've just stumbled upon Taylor Series on Wikipedia and I've been trying to obtain an expansion for $\arctan(x)$, but I can't manage to see a pattern for the $n$th derivative . Can someone come up with ...
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3answers
63 views

Taylor development of $\arctan(\cos(x))$ near $0$

How would I find the "Taylor development of $\arctan(cos(x))$ near $0$ at order $5$?" I am translating that from french, so I am not sure how I have to call it it english. By order $5$ I mean that I ...
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0answers
27 views

What is the radius of convergence of the derivative of a smooth Taylor series?

On this website I found that the derivative of a Taylor series has the same radius of convergence as the Taylor series itselves. However, there is no reference added, and I seem to be unable to find ...
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0answers
26 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
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39 views

Taylor expansion of matrix function

Find the Taylor expansion of the following expression as a function of $C$ around $C_0$ $ GC(I+GC)^{-1}$ in which $G$ and $C$ are matrices of compatible dimensions.
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27 views

Taylor theorem and a $C^{3}$ function with the following property…

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^3$ with $$f(a+h)=f(a)+f'\left(a+\dfrac{1}{2}h\right)h$$ whenever $a \in \mathbb{R}$ and $h \geq 0$. By applying Taylors Theorem to $f$ and to ...
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0answers
25 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 - ...
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2answers
44 views

Taylor Polynomials — $\cos(x)$ [closed]

Show that $$\forall x : \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots+\frac{(-1)^nx^{2n}}{(2n)!}$$ I know that this is true because it is one of the most common Taylor polynomials. ...
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0answers
37 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 ...
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2answers
45 views

Taylor series of a division-by-zero equation

I need to calculate taylor series of $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ at $t = 0$ to calculate limit $(\frac1{t^3}+\frac3{t^2})^{1/3} - ...
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1answer
44 views

Error Estimation Using Taylor's Theorem

I missed the lecture on this and was wondering if someone could explain the steps involved with this problem. I think that what I have to do is evaluate the polynomial up to the second derivative ...
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2answers
50 views

Find the Taylor series generated by f at x=a.

$f(x) = \frac 1 {9 - x}, a = 3$. The answer in the book is $$\sum_{n = 0}^{\infty} \frac{(x - 3)^n}{6^{n + 1}}$$but I'm not sure how to get the above.
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1answer
34 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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2answers
120 views

Remainder form of Taylor polynomial at $x_0$: $ \frac{1}{n!}f^{(n)}(x_0 +\theta(x-x_0))(x-x_0)^n$ with $\theta \to \frac{1}{n+1}$ as $ x \to x_0$

If the function $f: \mathbb R \to \mathbb R$ is $n+1$ times differentiable at $x_0$ and $f^{(n+1)}(x_0) \neq 0$, then a form of the remainder in Taylor's Formula is supposedly $$r_n(x_0;x) = ...
0
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1answer
31 views

How is the Harris Corner detector derived from a Taylor Expansion?

It looks more like an assumption about local linearity of the function when the shifts u and v are small. This description of ...
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3answers
52 views

Convergence of Taylor Series

My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states, "we will ...