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 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
74 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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55 views

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$

Compute the sum $\sum_{i=1}^{\infty}\frac{x^ {3i}}{(3i)!}$. I have no idea to find this sum. Can anyone give me a hint? Thank you in advance !
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1answer
45 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
40 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. ...
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1answer
678 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
12k 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
23 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 ...
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0answers
38 views

Power series expansion of $z\mapsto \frac{1}{1+z^2}$ around arbitrary point $x\in \mathbb R$

Determine the power series expansion of $z\mapsto \frac{1}{1+z^2}$ around $x\in\mathbb R$ with the respective radius of convergence. At first I tried working with Cauchy's integral formula to ...
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2answers
259 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
35 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
12 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 ...
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1answer
27 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
21 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
24 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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0answers
29 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 ...
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1answer
34 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
24 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
41 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
20 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: ...
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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
33 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) = ...
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0answers
22 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: ...
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1answer
46 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
121 views

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

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
31 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
100 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
18 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 ...
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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 ...
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1answer
95 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 ...
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3answers
145 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
31 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
49 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
49 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 ...
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1answer
66 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 = ...
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0answers
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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 ...
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0answers
24 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
507 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
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2answers
50 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
35 views

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
16 views

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) ...
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2answers
46 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
34 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 ...
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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: ...