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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What is the first order taylor expansion of a matrix to scalar function?

wikipedia says that if the real function f has the Taylor expansion: $f(x) = f(0) + f'(0) \cdot x + f''(0) \cdot \frac{x^2}{2!} + \dots$ then a matrix function can be defined by substituting x by a ...
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0answers
41 views

Taylor Series in any vector space?

I am working through Alexander Kirillov, Jr.'s An Introduction to Lie Groups and Lie Algebras, and on page 29 he does something I find puzzling. He claims that, since the exponential map is a local ...
2
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2answers
86 views

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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1answer
52 views

Understanding an apparent contradiction from naively applying Taylor's theorem and Fubini's theorem together

Suppose $f(x)$ is a bounded, $\mathcal{C}^{\infty}$ function on $\mathbb{R}$ for which the integral $$I = \int_{0}^{\infty}f(x)\ dx,$$ exists. Taylor's theorem implies $f$ admits a MacLaurin expansion ...
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2answers
41 views

Formula for the general term of the Taylor series of $\tan(x)$ at $x = 0$

Today we were taught different expansions; one of them was the series expansion of $\tan(x)$, $$\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15} + \cdots .$$ So, with curiosity, I asked my sir about next term. ...
3
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1answer
699 views

How to express the whole part $\lfloor x \rfloor$ as analytical function or Taylor/Fourier series?

And how to express $\{ x \} = x - \lfloor x \rfloor$ as function of $sin(x)$ and $sign(x)$?
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6answers
13k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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0answers
31 views

Power series expansion of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ and $z\mapsto \tan z$

Determine the power series expansion and radius of convergence of $z\mapsto \frac{\mathrm e^{z}}{1-tz}$ around $0$ with $t\in\mathbb C$. Determine the radius of convergence and the first three non-...
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2answers
311 views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
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0answers
40 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series (...
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1answer
13 views

Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
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1answer
30 views

Find limit using Maclaurin power series

I encountered the following problem: $$ \lim_{x\to 0} \frac{x-\ln(1+x)}{x-\arctan x} $$ I expanded $ \arctan x $ in the denominator up to the fifth term and get the following: $$ x - \left(x - \...
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0answers
23 views

Maclaurin series remainder in Ordo-form

I encountered a problem where two of the terms are the following: $$ \cdots+\frac{1}{2!}(x-\frac{x^3}{3!} + \omicron(x^5))^2 + \frac{1}{3!}(x-\frac{x^3}{3!} + \omicron(x^5))^3 $$ It's suggested that ...
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1answer
26 views

Macularian series for natural log

So, I know that $$ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... $$ Am I right in assuming that I can derive to follow by a subtitution of $-x$ $$ln(1-x) = -x - \frac{x^2}{2} - \...
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0answers
20 views

The “overdamped” approximation: deleting the higher order term in an ODE

Say we have an ode of the form $$ \epsilon \ddot{x} + a\dot{x} + b x = 0 $$ If $\epsilon$ is small enough the approximation $$ a\dot{x} + b x = 0 $$ is often done in physics; in fact, I'm interested ...
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36 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp 1,...
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1answer
38 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
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1answer
28 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
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1answer
45 views

Approximate integral using Taylor Series

I have to approximate this integral with an error lesser than 0.1 using Taylor Series. This is the integral: $$\int_0^1 \arctan(\frac{1}{x^{10}}) dx$$ If I understood, I have to determinate the Taylor ...
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1answer
21 views

Limit of a function containing $\Psi(x)$

The taylor series expansion of the function $$f(x)=\ln(1+x)$$ around zero is: $$f(x)=\sum_{k=1}^\infty\dfrac{(-1)^{(k+1)}}{k}x^k$$ Putting $x=1$ we have the alternating series: $f(1)=1-\dfrac{1}{2}+\...
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2answers
30 views

Taylor expansion of at a point different from $0$: should the variable be changed?

Find the Taylor expansion of $\arcsin x$ at point $1$. Can we change variable to get the series at point $0$? If yes how, and when do we change again to get back to $1$? More generally Let's ...
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3answers
34 views

How to differentiate the Taylor expansion?

We know the Taylor expansion of $f(x)$ at $a$ is and let it be $g(x)$, then $$g(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\ldots$$ My question is, is it ...
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2answers
23 views

First two terms of the Taylor series of the $n$-th iterated of a holomorpic function

Let $G$ be a region in $\mathbb{C}$ (i.e. $G ≠ \emptyset$ is simply connected and open), with $0 \in G$. Let $f: G \to G$ be a holomorphic function that's Taylor series (around $0$) has the shape $z + ...
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1answer
59 views

Explain inequality of integrals by taylor expansion

I try to understand why the following inequality holds. $$\left|\int_{|y|<1} e^{iuy}−1−iuy\ \, dy \right| \le \frac{1}{2} \cdot \int_{|y|<1} |uy|^2\ \, dy$$ Due to a hint I'm pretty sure, ...
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2answers
64 views

Induction Proof of Taylor Series Formula

I'm attempting to prove a formula for the taylor series of function from a differential equation. The equation is $$f(0)=1$$ $$f'(x) = 2xf(x)$$ I have found empirically that $$f(x) = \sum_{k=0}^{\...
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0answers
24 views

What is the general Taylor Expansion for the following function of a function.

guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: $$F(X(t+h))-F(X(t))=[X(t+h)-X(t)]\frac{dF}{dX}(X(t))+\\\frac{1}{2}[X(t+...
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1answer
50 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
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1answer
127 views

Power series expansion of $x\ln(\sqrt{4+x^2}-x)$ [closed]

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$. I know that I must find power series expansion of $\ln(\sqrt{4+x^2})$ but it doesn't help. Can anyone give me a hint? many ...
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1answer
32 views

Logarithmic Taylor series question [closed]

Consider the transformation of variables, x = $\frac{y+1}{y-1}$ How would you develop log(x) as a Taylor Series in y about zero?
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3answers
104 views

How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?

Can anyone help me with the above inequality? I tried looking at the series expansion and I guess the answer indeed lies there, but I fail to see it. Thanks
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0answers
19 views

Derive orthogonal transformation

Let $R$ and $R'$ be two cartesian co-ordinate systems and $\phi=(\phi_1,\phi_2,\phi_3):\mathbb{R}^3\to\mathbb{R}^3$ a map that relates the $\textbf{x}=(x_1,x_2,x_3)$ co-ordinates of $R$ with the $\phi(...
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0answers
26 views

How do we know that a function can be written as a power series?

Most proofs of a Taylor series or a Maclaurin series assume that the function can be written as a power series. If a function can be written as a power series then: $$f(x)=\sum_{n=0}^\infty \frac{f^{(...
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1answer
97 views

Proving analyticity of an integral function over $\mathbb{R}^{n}$

Let $U\subsetneqq\mathbb{R}^{n}$ be open, $\varepsilon>0$ and consider the function $$f_{\varepsilon}(x)=\frac{\pi^{-\frac{n}{2}}}{\varepsilon^{n}}\int_{U}\exp\left\{-\left\|\frac{x-y}{\varepsilon}\...
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3answers
150 views

Solving the second taylor polynomial

So I've found myself in a predicament when trying to implement the second Taylor polynomial. Here is my question: Let $f(x) = \sqrt{x}$, find the second Taylor polynomial $P_2(x)$ for this ...
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1answer
33 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
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1answer
50 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...
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1answer
50 views

Find the taylor expansion to $(x^2 + x)e^{2x}$

My task is this: Find the taylor expansion to$$f(x)=(x^2 + x)e^{2x}.$$ My work so far: We should get $$e^{x}=\sum_{n=0}^\infty\frac{x^n}{n!}\implies e^{2x}=\sum_{n=0}^\infty\frac{(2x)^n}{n!}\...
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1answer
69 views

Finding taylor expansion of $\cos^2x$ and $\sin^2x$

My task is this: Find the taylor-series of $\cos^2x$ and $\sin^2x$. My work so far: We know that $\cos^2x \backslash \sin^2x = \frac{1\pm \cos 2x}{2}$, and the series for $\cos x = \sum_{n=0}^\...
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0answers
18 views

What is the name of this approximation?

I remember studying a while back about an approximation method where the error is calculated using $$ E_{n}=M_{n+1}-a_{n+1} \widetilde{T}_{n+1} $$ Where $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
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1answer
28 views

A basic question about the decay rate of $te^{-t}$ as $t$ tends to infinity

It is well-known that $te^{-t}$ tends to $0$ as $t$ tends to infinity. But I want to know the decay rate of $te^{-t}$ as $t$ tends to infinity. Using Taylor expansion of $e^{t}$ we have: $${t /e^{t}}=...
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0answers
25 views

Range of values of $x$ for which the expansion $\ln(2+x)$ to valid

It is known that $$\ln(1+x) = \sum_{n=1}^\infty{(-1)^{n+1} \dfrac{x^n}{n}}$$ for $-1<x\leq1$. Question: What is the range of values of $x$ for which the expansion of $\ln(2+x)$ is valid? I ...
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2answers
52 views

How do you show that a power series of $e^x$ at a point converges to $e^x$?

My task is this: Find Taylor-expansion of $e^x$ at the point $1$, the convergence interval $I$ and then show that the series converges to $e^x$. My work so far: By using $Te^x @ x= 1$ we should get ...
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1answer
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Taylor expansion of function [closed]

I try to figure out how the taylor expansion of the following function looks like, but so far I wasn't successfull: $z↦e^{iuz}−1−iuz$ for $|z|<1$. Who has an idea?
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0answers
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Taylor series approximation of inverse trigonometric function

Suppose we have a function of three variables $a,b,c$ defined as, $f(x,y,z)=\arctan\left(\frac{\sqrt{x^2y^2-z^2}}{y^2-z}\right)$. Suppose $x=a, y=b, z=c$ satisfy the following property: (1) $a,b,c>...
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2answers
48 views

Find the fourth Taylor polynomial of f(x)=ln(x+1) at x=1

Let $f(x)=\ln(x+1)$ then (a) find the fourth Taylor polynomial of f at x=1 and (b) use part (a) find the approximate the value of ln(2.2) correct 4 decimal (c) Find an estimate for the error in ...
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2answers
35 views

Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
1
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1answer
33 views

How to expand $x^n$ as $n \to 0$?

I am trying to expand $x^n$ in small $n$ using Taylor series. Using wolfram alpha, I found that it is $1+ n\log(x) + \cdots$ I tried to Taylor expand $x^n$ around $n=0$ but I cannot get this result.
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1answer
45 views

Interesting behavior of the expansion of $_1F_2(\alpha/2;3/2,\alpha/2+1;y^2/4)$ near $y=\infty$

When we use Mathematica 10.0 to expand generalized hypergeometric function $_1F_2(\alpha/2;3/2,1+\alpha/2;y^2/4)$ near $y=\infty$ with $\alpha$ a complex number, we obtain: $${_1F_2}(\alpha/2;3/2,1+\...
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6answers
906 views

Why the existence of Taylor series doesn't imply it coverges to the original function

Please note that I've read this question and it did not address mine. I've been presented with the following argument regarding Taylor series: We have a function $f(x)$, now assume that there ...
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1answer
23 views

Analytic and smooth functions

In my work, I first make an assumption: Assume the function $f(x)$ is an analytic function of $x$. Based on this assumption, I expand $f$ as Taylor series $$ f(x)=f_0+f_1x+f_2x^2+f_3x^3+\dots $$ ...