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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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Taylor series expnasion

I was given the following question : Evaluate the series expansion of $f(x) = \ln(\frac{1-x}{1+x^2})$ and determine the radius of convergence. So my first move was to split the logarithm into $\ln(1-x)...
Johann Carl Friedrich Gauß's user avatar
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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form

I came across this limit problem: $\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$ Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
Afsheen's user avatar
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+100

'deducing' a bound using the first order taylor series. How to make it more precise?

So, I just saw a ‘proof’ that the generalized birthday problem has a median of C*sqrt(n). Though the probability in question is interesting, this question is more about calculus and maybe asymptotics ...
josinalvo's user avatar
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Construct/prove existence of a function with given expansions at two different points

Consider two non-constant real polynomials $f(x)$ and $g(x)$: $$f=f_0 + f_1 (x-x_0) +...+f_N(x-x_0)^N $$ $$g=g_0 + g_1(x-x_1) +...+g_M(x-x_1)^M $$ where $f_0...f_N,g_0...g_M,x,x_0,x_1 \in \mathbb{R}$ ...
Quillo's user avatar
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How to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$

In the calculation of a problem, I need to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$ holds for any positive integer $n$. I got the expansion ...
Jun Wang's user avatar
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Taylor expansion of $\int_{0}^{\omega_0}\frac{\sin\left(\frac{2N+1}{2}\omega\right)\cos(\omega n)}{\sin\left(\omega/2\right)}d\omega$ and similar

I would like to compute the expansion of the following integrals near $N = + \infty$ up to $\mathcal{O}(1/N^2)$: $$ \int_{0}^{\omega_0} \frac{\sin \left( \frac{2 N + 1}{2} \omega \right) \cos(\omega n)...
Francesco Orso Pancaldi's user avatar
1 vote
1 answer
40 views

How do we determine what small angle and small $x$ are for a simple pendulum to justify linear approximation?

Consider a simple pendulum consisting of a point-like mass $m$ attached to a massless string of length $L$ from a fixed support and constrained to move in a vertical plane. Here is a picture of this ...
xoux's user avatar
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2 votes
3 answers
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$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ when $n$ is even and $x\geq 0$

Let $n\geq 2$ be an even integer and $x\geq 0$. I want to show that $$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}.$$ Assume first that $x\in (0,\pi]$. By Taylor's theorem with Lagrange ...
brised by Linear Algebra's user avatar
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1 answer
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Do you need L'Hôpital's rule to prove Taylor's formula?

I recently read a Quora answer. The answerer was asked to solve the limit $$\lim_{x\to0}\frac{\cos x-e^x}{\sin x}$$ without using L'Hôpital's rule. The answerer used the Taylor series expansion of the ...
Elvis's user avatar
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1 answer
81 views

Find the Laurent expansion of $f(z) = \sin{\frac{1}{z(z-1)}}$ in $0<|z-1|<1$

Here is my idea: $\sin{\frac{1}{z(z-1)}} = \sin{\left( \frac{-1}{z} + \frac{1}{z-1}\right)} = \sin{\left(\frac{1}{z-1} - \frac{1}{z}\right)} = \sin{\frac{1}{z-1}}\cos{\frac{1}{z}} - \cos{\frac{1}{z-1}}...
Irbin B.'s user avatar
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summation of a series on a lattice

I am a physics student. And I am working on some wave function on some lattice. One question I encountered in my study is studying the following summation $$f(x)=\sum_{m,n \in Z}\cos((m-n)gx)\exp(-\...
Tixuan Tan's user avatar
2 votes
1 answer
52 views

Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem

RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$ I am trying to prove this specifically using Taylor theorem. Here is what I have so far: $\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
punypaw's user avatar
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4 votes
3 answers
151 views

Find $\sum_{n=1}^{\infty}\left(n\sin\left(\frac{\pi n}{2}\right)\left(e^x-1-\frac{x}{1!}-\frac{x^2}{2!}-\cdots-\frac{x^n}{n!}\right)\right)$

Find the value of $A$ where $$A=\sum_{n=1}^{\infty}\left(n\sin\left(\frac{\pi n}{2}\right)\left(e^x-1-\frac{x}{1!}-\frac{x^2}{2!}-\cdots-\frac{x^n}{n!}\right)\right)$$ By using the expansion of $e^x$,...
MathStackexchangeIsMarvellous's user avatar
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1 answer
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Taylor expansion of the function $f(x)=(1+x) ^ {1/x}$ in $x = 0$, to the second order

I there, I have to compute the taylor expansion of this function: $$f(x) = (1+x)^{1/x}$$ in $x = 0$, to the second order. First I have transformed the function: $$(1+x)^{1/x} = e^{\frac{\ln(1+x)}{x}}$$...
Bmb58's user avatar
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Series expansion involving integrals of higher derivatives and Bernoulli polynomials

Given a (smooth) function f defined on $[0, 1]$, I am looking for a series expansion of the form $$ f(x) = \sum_{n = 0}^\infty \frac{c_n}{n!} P_n(x), $$ where $c_n = \int_{0}^{1} f^{(n)}(t) dt$, where ...
Maxime Lucas's user avatar
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Expand $\sqrt{1-4x}$ into an infinite power series

I'm reading the book Math Girls. At one time (p. 131), a closed form was obtained for generating function $C(x)$ like below. $$C(x)=\frac{1-\sqrt{1-4x}}{2x}$$ To facilitate further deduction, the ...
Lingxi's user avatar
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1 answer
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Finding inverse to inner automorphism (Humphreys' Lie Algebra book)

This is basically an algebra question. In Humphreys' book on Lie algebras he states that one can find the inverse to $\exp \delta$, where $\delta$ is a nilpotent derivation - say $\delta^k=0$, by ...
raynea's user avatar
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2 votes
2 answers
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How do I get the positive series expansion of $(e^{\alpha x}-1)^{-1}$?

I couldn't figure out how the second line came out in the expression below. I basically have to expand $(e^{\alpha x}-1)^{-1}$ into a series, where $\alpha > 0$. My first try was $\beta \equiv e^{\...
xiver77's user avatar
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Multivariable taylor - why does $R_k(a) = o(||h||^k)$

Let $f : E \to \mathbb{R}$ be $C^k$ for $E \subset \mathbb{R}^n$ open. Choose $a \in E$ and $h \in \mathbb{R}^n$ such that $a + h \in E$. Then $\exists \theta \in (0,1)$ such that $f(a + h) = f(a) + (...
Camiel's user avatar
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1 answer
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exact value of this series expansion and integral identity

is there a closed value for the series $$ \sum_{n=0}^{\infty}\frac{(-x)^n}{(n!)^2}=G(x) $$ if so is this integral identity correct ? $$ e^{-x} =\int_{0}^{\infty}G(xt)e^{-t} dt $$ i just have expanded ...
Jose Perez's user avatar
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Finding twice-differentiable $f$ such that $f(x) = x^{-a} - 1 + f(1) + f'(1)(x-1)$

$a$ is a constant, $x \in (0,\infty)$. I encountered this differential equation playing around with some material, unsure if it hides something interesting. In fact, plugging $f(1)$ returns an ...
Lele's user avatar
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4 votes
1 answer
83 views

Formula for a matrix's characteristic polynomial as the exponential of a series in $\mathrm{tr}(A^k)$

I want to prove the following equation $$ \det(x - A) = x^n \exp \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k x^k} \mathrm{tr} (A^k) $$ where $A \in \mathbb C ^{n \times n}$. In the case where $A$ is a ...
notmyrealname's user avatar
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2 answers
61 views

Use Taylor expansions to find if an improper integral converges

I'm trying to decide if the following integral $$\begin{align*} \int _{1}^{\infty} \frac{(e^{1/x^{2}}-1)^{\alpha}}{\log^{\beta}\left( 1+ \frac{1}{x} \right)} \ dx \end{align*}$$ is convergent or not. ...
M_k's user avatar
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1 answer
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Finding the taylor expansion of $f(z) = \frac{1} {1-z-z^2}$ at $z=0$ [duplicate]

Consider the function $f(z) = \dfrac{1} {1-z-z^2}$. Defined over the complex numbers. Find the Taylor expansion of $f$ at $z=0$. Finding each order derivative at $0$ is a lengthy and time-consuming ...
Alp1091's user avatar
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1 vote
1 answer
110 views

Can a function be strongly differentiable but not continuously differentiable?

A similar question was asked before (however, there were a few issues with the definitions and answer given, as I pointed out over there): Can a function be differentiable but not strongly ...
William M.'s user avatar
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0 votes
1 answer
28 views

Finding Expectation of a Uniform random variable from its moment generating function

From Taylor series, if I need to get the k-th moment, I need to find the k-th derivative of the Moment Generating function. If I have $X \sim \text{Uniform}(0,1)$, the MGF $M_{X}(s)$ is; $$M_{X}(s) = \...
moseskabungo's user avatar
1 vote
0 answers
27 views

Notation Clarification for Taylor's Theorem in Higher Dimensions

The following lemma is stated in the coursebook for a paper I am taking on analysis in higher dimensions. (The Lemma is working towards proving Taylor's theorem in higher dimensions). I am having ...
raynerk's user avatar
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0 answers
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Finding $\lim\limits_{x\to0} \frac{x^3f(\mathbf{u})}{f(x)\sin^3 \mathbf{u}}$ via Taylor remainder formula

Assume that the second derivative of the function $f(x)$ is continuous, and $f''(x)>0$, $f(0)=0$, $f'(0)=0$, find the limit: $\lim_{x \to 0} \frac{x^3f(u)}{f(x)\sin^3u}$, where $\mathbf{u}$ is the ...
xin zen's user avatar
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1 vote
2 answers
50 views

Examples of expansions of the exponential of a sum of two matrices [closed]

The exponential function of a matrix is fundamental in mathematics, physics and beyond. One can define it using the power series $$ e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}  $$ For any matrix $M$ ...
Frederik Ravn Klausen's user avatar
3 votes
1 answer
69 views

The Taylor polynomial of the function $f:\mathbb{R^2} \rightarrow \mathbb{R}$ at $(x_0,y_0)=(3,4)$ is $\ T_2(x,y)=(x-7)^2 - xy +6y^2$

The Taylor polynomial of the function $f:\mathbb{R^2} \rightarrow \mathbb{R}$ at $(x_0,y_0)=(3,4)$ is $\ T_2(x,y)=(x-7)^2 - xy +6y^2$ Find the hessian matrix $H_f(3,4)$ and the gradient $\nabla f(3,4)$...
Dary 's user avatar
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1 vote
1 answer
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Maclaurin series for $(1+e^x)^{-1}$ and radius of convergence

I am looking at some exercises in a book. In one of the exercises, we are asked to give the first three terms of the series for $(1+e^x)^{-1}$ and state which values of $x$ the series is convergent. I ...
hhh3's user avatar
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1 vote
2 answers
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Error Caused By Taylor Series Approximation In A Mechanics Problem

I'm currently going through "An Introduction to Mechanics" by Kleppner and Kolenkow. On pages 36 - 37, Kleppner discusses how to find an approximate solution to a Physics problem. I have a ...
Ethan Chan's user avatar
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2 votes
2 answers
83 views

Finding the natural boundary of $\sum_{n=0}^{\infty}\frac{z^n}{n^k}$ for $k \ge 2$

Let $k\ge 2$, given $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^k}$$ It is easy to see that it converges for $|z|\le 1$, but how can it be analytically continued beyond the unit circle? Hadamard proved ...
kmxzc's user avatar
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3 votes
2 answers
217 views

Power series representation of $1/(2+x)$

Why the power series representation of $\frac{1}{2+x}$ is $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n+1}} x^n$$ and not $$\sum_{n=0}^{\infty}(-1-x)^n$$ based on $\frac{1}{1-x}$ = $\sum_{n=0}^{\infty}x^n$ ...
Darren's user avatar
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4 votes
1 answer
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Can I apply taylor expansion to exponents?

If I have a limit in this form: $$\lim_{x\to x_0}f(x)^{g(x)}$$ can I expand $f(x)$ and $g(x)$ to end up with a limit like: $$\lim_{x\to x_0}\left(f(0) + f'(0)x + \frac{f''(0)}2x^2 + \cdots\right)^{g(0)...
Elvis's user avatar
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0 votes
1 answer
30 views

Taylor expansion for a differential operator

I'm struggling with a problem in which I need to expand a differential operator near a known function. The problem is the following: $L(u)$ is a differential operator, in my problem: $L(u)$ is defined ...
Luca Javier Gomez Bachar's user avatar
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1 answer
55 views

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{\cos(\sqrt{xy})-1}{y}$ using Taylor series

To evaluate the following limit $$\lim_{(x,y)\to(0,0)}\frac{\cos(\sqrt{xy})-1}{y}$$ I tried to use the fact $$\sqrt{xy}\leq\frac{x+y}{2}$$ but it didn't work as I thought. My friend said I could use ...
mvfs314's user avatar
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1 vote
1 answer
46 views

Taylor-Laurent series expansions

I'm having some issues finding how to series expand some complex functions that my professor gave past years in exams. For example, in this exercise, it is asked to find the first two terms of the ...
deomanu01's user avatar
  • 113
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0 answers
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How to expand a complex function around the point at infinity?

I came across a problem that asked to expand the function $$ f(z) = \frac{1-e^{2iz}}{z^2} $$ both around the point $z=0$ and $z=\infty$. The correct expansion around $z=0$ should be $$ f(z) = -\sum_{k=...
deomanu01's user avatar
  • 113
0 votes
1 answer
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Coordinate-free definition of smooth map

Say $(\mathcal E, V)$ and $(\mathcal F, W)$ are Euclidean spaces. A map $F : \mathcal E \to \mathcal F$ is said to be differentiable at $p_0 \in \mathcal E$ if there exists a linear map $L : V \to W$ ...
markusas's user avatar
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1 vote
1 answer
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Asymptotics for limits of functions in many variable

A stupid question perhaps. When dealing with limits of functions in many variables, can I use asymptotic estimations? For example, can I just say $\sin(xy) \sim xy$ as $(x, y) \to (0, 0)$ or is it a ...
Heidegger's user avatar
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1 answer
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Taylor Expansion of a vector-valued function with 2 vectors as input

Let a function $f(x,u): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$. I wonder how to expand it around $(x_n, u_n)$. For the time being, keeping it only up to the first order is enough ...
Marios Stamatopoulos's user avatar
1 vote
0 answers
64 views

Having trouble proving Taylor Expansion

In the book Introduction to Manifolds by Loring W. Tu, the author introduces a method for obtaining the Taylor expansion on "star-shaped" (i.e., any line segment connecting points in the set ...
PauseAndPonder's user avatar
2 votes
1 answer
35 views

Higher order Frechet derivatives viewed as bilinear maps, on Taylors theorem

So I have been studying some introductory non-linear analysis. I am currently looking at higher order Frechet derivatives and I want to proof-check/ make sure I got something right. So given $X,Y$ ...
Bigalos's user avatar
  • 394
3 votes
1 answer
84 views

Taylor expansion of $x^2/(k+x^2)$

I'm trying to write down a formula for the Taylor expansion of $x^2/(k+x^2)$ around $x=z$. The Taylor expansion for $x/(k+x)$ around $x=z$ is \begin{align*} \frac{x}{k+x} = \frac{z}{k+z} + k \sum_{j=1}...
tomjonson's user avatar
1 vote
2 answers
66 views

Euler's Gamma function and convergence of an integral

I'm trying to calculate the following integral: $$\int_0^\infty e^{-x}\cos(x)dx$$ and I figured a way to do it would be to expand the cosine in its Taylor series, switch the integral and summation ...
Lagrangiano's user avatar
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0 answers
55 views

Show that the function $f(x)=\begin{cases} e^{-1/x^2} & x\neq 0 \\ 0 & x=0 \end{cases}$ has no Maclaurin's series [duplicate]

Show that the function $$f(x)=\begin{cases} e^{-1/x^2} & x\neq 0, \\ 0 & x=0 \end{cases}$$ has no Maclaurin's series expansion but the function has derivative of all order everywhere. We have $...
user1942348's user avatar
  • 3,903
1 vote
0 answers
30 views

Asymptotic expansion with singularity subtraction

I am studying a set of lecture notes on the asymptotic expansion of integrals in the case of a removable singularity. While filling in some missing details and fixing some typos, I encountered some ...
123prior's user avatar
10 votes
1 answer
146 views

$f:\mathbb R\to [0,\infty)$ is a 3 times differentiable and $\max_{x\in\mathbb R}|f'''(x)|\le 1$. Prove that: $f''(x)+\sqrt[3]{\frac{3}{2}f(x)}\ge0 $

I have a problem in Analysis: Let $f:(-\infty,\infty)\to [0,\infty)$ be a three times differentiable and satisfy: $$\max_{x\in\mathbb R}|f'''(x)|\le 1$$ Prove that: $$f''(x)+\sqrt[3]{\frac{3}{2}f(x)}\...
Đạt Nguyễn's user avatar
1 vote
0 answers
32 views

Taylor Approximation of Third Order

Problem: Given the function $ f(x, y) = e^{x^2} \log(1 + x + y) $ near the point $ (0, 0) $, find the third-order Taylor approximation of the function at $ (0, 0) $ using known series and verify the ...
j.primus's user avatar

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