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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Comparing Two Multilinear Polynomials based on Multivariable Taylor Expansion

Given two linear functions $f(x)$ and $g(x)$ defined on real values, let's say that I want to show that $f(x) > g(x)$ for all real $x > t > 0$. According to the order-1 Taylor expansion at ...
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2answers
66 views

A Taylor series expansion of $e^{ix}$

In Probability Theory by Athreya and Lahiri, they give a very elegant proof of Central Limit Theorem (The Lindeberg one) wherein they use a lemma: For $x \in \mathbb{R}$ and $r \geq 1$, ...
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0answers
24 views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: ...
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32 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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2answers
35 views

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence.

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to ...
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1answer
88 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
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1answer
31 views

Given that $f(1) = f'(1) = 1$, use Taylor polynomials to show that $\lvert f(x) - x \rvert \leq A(x - 1)^2$

Given that $\ f$ has continuous second derivatives in$\ [0,2]$ and $\ f(1)=f'(1)=1$, I'm trying to prove that for every $\ x \in [0,2]$ exists an A so that: $$ |f(x)-x| \le A(x-1)^2 $$ The second ...
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70 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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3answers
32 views

Proof:Taylor expansion of inverse trigonometric functions

I find it quite difficult to remember the Taylor expansion of inverse trigonometric functions.Actually in school we have been just taught the series (for finding limits in calculus without teaching us ...
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44 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
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3answers
96 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
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1answer
41 views

Prove that for every $x$ in the area of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$

I need to prove 2 things: Prove that for every $x$ in a neighbourhood of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}.$ What I did is that I calculated the derivatives of ...
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5answers
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Why do un-integrable funcitons exist?

By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. I recently learnt that any differentiable function can be expanded using the Taylor ...
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1answer
66 views

Determine the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$

I need to calculate the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$ . It seems very "similar" to Taylor expansion of functions arcsin(x) and its derivative for x = -2. It is known: ...
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1answer
27 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
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1answer
20 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
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16 views

Using Taylor series of function f1(x) in Taylor series of function f2(x) defined in open discs D1 and D2 when D2 lies inside D1

We have two open discs, D1 and D2, whose centres are C1 and C2 respectively. The Taylor series of function f1(x) is defined in open disc D1 while the Taylor series of function f2(x) is defined in open ...
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5answers
67 views

$\lim_{x \to 0} \cfrac{e^{2x} - \ln(1-x) - \sin(x)}{\cos(x)-1}$ using Taylor Expansions

As a preface- a very similar question is here: Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$ But, my actual question is, when we substitute the ...
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1answer
25 views

error term for double integral approximation by midpoint rule

I found following statement in the book that I'm reading: Using Taylor series expansions it is easy to prove that: $$ \left|h^2\cdot ...
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2answers
77 views

Expand the Taylor series for the following mind-boggling expression at $x = 0$

Mind-boggling expression is: $$f(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)}$$ I started by using the quotient rule and expanding the denominator terms in the hopes of finding some pattern ...
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1answer
61 views

How to solve this limit using laurent series?

$$\lim_{x\to\infty}\left(\left(\frac{x^2+5}{x+5}\right)^{3.7}+\left(\frac{x^3+5}{x+5}\right)^{1.6}\right)^{20/37}-\left(\left(x-5\right)^{3.7}+(x^2-5x+25)^{1.6}\right)^{20/37}=60$$ It is possible to ...
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1answer
31 views

Proving that a function is increasing

I have this problem Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, such that its Taylor series convergers to function $f$ everywhere. For every derivative of the function $f$ we have that ...
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1answer
28 views

Computing first three non-zero terms of a Taylor series

I have a function $F(t)=\int_0^t \sqrt{1-x^8} dx.$ I have to find the first three non-zero terms of a Taylor series of $F$ around the point $a=0.$ Since I want the Taylor series I started with the ...
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1answer
33 views

Does an inequality between first-order Taylor approximations imply the same for the functions?

Assume that $f$ and $h$ are functions from $\mathbf{R}^n$ to $\mathbf{R}^1$ and continuously differentiable. Also assume that $f(z)=h(z)$ at some point $z \in \mathbf{R}^n$. Could we then show that ...
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2answers
35 views

Extending a bounded holomorphic function past its boundary

Suppose I have a bounded holomorphic function on the unit disc, centred at the origin. Can I always extend this beyond the origin to say a disc of radius $1 + \epsilon$ for some $\epsilon > 0$? My ...
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1answer
16 views

Taylor expansion of $f$ in stability analysis of 2-step Adams-Bashforth method

Given the two-step Adams-Bashforth method $$ u_{n+1} = u_n + \tfrac{h}{2}(3f_n - f_{n-1}) $$ find its order. Some notation: $t_n = t_0 + nh$ is the $n$-th node and $y_n = y(t_n)$; $f_n$ ...
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0answers
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Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
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1answer
17 views

Index confusion in Tu's treatment of Taylor's Theorem with remainder in “An Introduction to Manifolds”

In Tu's book, specifically the section on "Taylor's Theorem with remainder", there appears to be a changing of the meaning of some subscripts which isn't noted. The theorem states that if $f$ is a ...
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1answer
37 views

Taylor expansion at discontinuous point

a) Find the Maclaurin expansion of the following function: $$f(x)=\int\limits_0^x \frac{1-e^{-t^3}}{t^2} \mathrm{d}t$$ end b) evaluate the $ \displaystyle \lim_{x \to 0^{+}} f^{(29)}\, (x) $ The ...
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1answer
35 views

approximation of $\pi$ by $\arctan$

Determinate the order n of the Maclaurin polynomial for $f(x)=4tan^{-1}x$ so that the remainader term $|R_{n}(1)|<0.000005$. Here $R_{n}(1)=\frac{f^{(n+1)}(c)}{(n+1)!}$ for some c between 0 and 1 ...
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1answer
93 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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31 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
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81 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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1answer
13 views

Use the remainder estimation theorem to find the maximum value of error?

My answer for part (bi) was $\frac{x}{2}$ - $\frac{x^2}{4}$ My attempt for part b(ii) was to find $\frac{g^{'''}(z)}{3!}$($x^3$) = $\frac{8}{3!(2+2z)^3}$ where z ∈ [0, $\frac{1}{2}$]. To find the ...
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1answer
20 views

The Maclaurin series and taylors theorem for $\sinh(2x)$

I am currently studying for an exam next week but am struggling to the second part of this question. I have figure out the Maclaurin series for $\sinh(2x)$, however am unsure how to estimate the ...
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1answer
19 views

Calculate the 3rd order Taylor polynomial about $x=1$?

Calculate the $3$rd order Taylor polynomial about $x=1$ for the function $f:[-3:\infty) \longrightarrow \mathbb{R}$ given by $f(x)=\sqrt{x+3}$. I know that the formula for the Taylor Polynomial ...
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1answer
32 views

Calculate the Taylor series of $\sin$ around $3$

This should be simple but I'm having trouble with it. So by definition the series looks like $$\sin 3 + \cos (3) (z-3) - \frac{\sin (3) (z-3)^2}{2!} - \frac{\cos (3) (z-3)^3}{3!}+...$$ To be able ...
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1answer
72 views

Approximation for probability of at least $t$ events

I'm reading through a paper, and they are discussing the approximate probability that $t+1$ out of $t^b$ events occur, where $b$ is a constant, and the probability of each event occurring is ...
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25 views

How do I expand this function?

Can anyone help me to expand this function: $$ f(\theta) = R \epsilon_0 \left(\frac{\ln{ (1+L/d \tan{( \theta )})}}{\tan{(\theta)}}\right)$$ I want to expand it for small $\theta$ (I guess around ...
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30 views

Determine the 19-th, 20-th and 21-st order Maclaurin polynomial?

The answers tell me that the solution is: Firstly, I don't understand the notation "x$\to$". What does this mean? This is my attempt to find the maclaurin polynomial for x(1-cos2$x^3$): ...
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2answers
109 views

Taylor series not converging, other example than $\exp(-1/x^2)$?

The usual example for non-converging Taylor series is $g(x) = \exp(-1/x^2) \; \forall x \neq 0, g(0) = 0$: the Taylor series around $x=0$ is zero, but $g$ isn't zero for any $x \neq 0$. What's not so ...
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1answer
59 views

Proof that taylor series converges to function using taylors inequality

I would like to proof that the function $f(x)=\frac{1}{\sqrt{1-x}}$ converges to its Maclaurin series $$Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n$$ for $0<x<1$ by using taylors ...
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2answers
23 views

Find the quadratic Taylor polynomial of erf about a=0?

The answers say that $P_3(x) = 1 + \frac{1}{2} x^2 $ I understand that this is the sum of the first four terms, however I don't know how they calculated this. I know that the formula for the ...
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1answer
70 views

Coefficient of operator and how to do it

This question stems from this $$ \frac{1}{x+z}- \frac{1}{x} = \sum_{k=0}^\infty \frac{z^k}{k!}\frac{d^k}{dx^k}[\frac{1}{x}] $$ Now, i need to find the Bell Polynomial of $\frac{1}{x}$, $$ ...
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3answers
59 views

How to expand $\sqrt{x^6+1}$ using Maclaurin's series

The expansion would be $\sum_{n=0}^\infty$$\frac{1}{2}\choose n $$x^{6n}$ How to evaluate binomial coefficient with rational numbers? If $\frac{1}{2}\choose n $=$2n\choose n $$\times ...
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1answer
26 views

Taylor polynomials: remainder formula for expansion around $\infty$.

By definition of Taylor polynomials, we have $$f(x)=f(x_0)+f'(x_0)(x-x_0)+...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+R_n(x,x_0)$$ where $R_n$ is the $n-$th remainder . Let $U(x_0)$ the neighborhood of ...
1
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1answer
45 views

Taylor expansion of fraction

I am trying to Taylor expand the function $f(x) = \frac{x}{x+3}\frac{1}{x-2}$ aound the point $x_0 = 2$. Clearly, the last factor explodes around this point, so I will try and expand that term. ...
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2answers
39 views

Proving the inequality $2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$ for $n\in\mathbb{N} $ and $x>0$

Prove that for all $n\in\mathbb{N}$ and $x>0$, $$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$ The last class was about Taylor polynomial of functions, so I ...
1
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2answers
42 views

Taylor series of $e^{i \sin z}$

How can I find the Taylor series of at $z=0$ (where $z$ complex ) of: $e^{i \sin z}$? What I wrote is: $$e^{i \sin z}= \sum_0^\infty \frac{(i \sin z)^n}{n!}$$ Is that right? And how it can be more ...
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0answers
13 views

linear term of $\sqrt{ 1 - A \exp^{2iwt} - B exp^{2iwt}}$ does it make nonsense to analysis of dynamics?

I have the following problem: I am building a Lagrange Euler equation near the position where some vector component equals to 1, while others no. I do the following substitutions: $$\vec x = ...