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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0
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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{...
2
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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 ...
0
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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 ...
-1
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0answers
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,...
0
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1answer
38 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
0
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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) ...
1
vote
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 ...
0
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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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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 ...
0
votes
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 + ...
1
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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+...
0
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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 ...
1
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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?
6
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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
0
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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(...
1
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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^{(...
1
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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). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
2
votes
2answers
87 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 ...
3
votes
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 ...
1
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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 ...
0
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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!}\...
1
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1answer
70 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}^\...
2
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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 $\widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}$, ...
0
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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}}=...
0
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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 ...
2
votes
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 ...
0
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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, ...
-1
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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?
1
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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}^{\...
0
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0answers
18 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) $a,b,c>...
0
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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 ...
1
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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
vote
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.
2
votes
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}\...
9
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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 ...
0
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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 $$ f(x)=f_0+f_1x+f_2x^2+f_3x^3+\dots $$ ...
9
votes
2answers
312 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 ...
0
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0answers
24 views

Solve the following ODE using a Maclaurin expansion of the non-linear terms

Find two proper series solutions about the ordinary point $x=0$ of $$y''+e^xy'-y=0.$$ My proposed solution: Note that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$ Assume there exists a power series ...
2
votes
1answer
21 views

Taylor and Macluarin series deriving

Hi to everyone Here i am studying Taylor series. $$f(x)=c_0 + c_1 (x-a) + c_2 (x-a)^2+ ...$$ $$ f(x)= f(a) + \frac{df(a)/dx}{1!}(x-a)^1 + \frac{d^2f(a)/dx^2}{2!}(x-a)^2 ...$$ Well my problem is ...
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0answers
15 views

Estimate an Taylor approximation II

i am doing some exercise for my numerical analysis course. And i found myself wondering if the following argument is legal. The context of this exercise is the smoothend newton algorithm, especially ...
0
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1answer
20 views

Laurent series about singular point for: $\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$

$\left( \frac{1+x}{1-ax}\right)^{\frac{1}{3}}$ I wish to find the Laurent series about the singular point $x=1/a$. I can find an expansion for the left side ($x=0$) and the right side ($x \rightarrow ...
1
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1answer
40 views

Series expansion of infinite series raised to the $n$th power

So I know there is a well-known straightforward way to expand something like $$(a+b)^n$$ and that there are formulas which allow us to expand trinomials and multinomials in general. My question is, ...
0
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1answer
22 views

Taylor Series of a composition of functions

I have to find a Maclaurin series of the following function: $y = D\sin(C\arctan(Bx - E(Bx - \arctan(Bx)))) + Sh$ I wasn't able to find it by hand. Thanks in advance!
0
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0answers
10 views

The remainder of taylor approximation, lagrange form of the remainder. The idea

We know that the formula for the remainder of taylor approximation is: $$R_n(x) = \frac{f(z)^{n+1} *(x-a)^{n+1}}{(n+1)!}$$ But also we have the formula: $$R_n(x) = \frac{M *(x-a)^{n+1}}{(n+1)!}$$ ...
1
vote
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+\...
0
votes
2answers
46 views

Find the Maclaurin series for $\cos^2(x)$

I am given this as a hint: $\cos^2(x) = \frac{1 + \cos(2x)}{2} \\$ I am not really sure how to start this one, would it just be the regular Maclaurin series squared. For example: $ (\sum_{n=0}^\...
0
votes
1answer
17 views

Relation between coefficients of two different power series.

Let $$f(z) = \sum_{n\geq 0} = a_nz^n, a_n\in\Bbb{C}$$ has a radius of convergence $\rho$. Then we can write $f(z) = \sum_{n\geq 0} b_n (z-\frac{\rho}{2})^n$ for $\{z: |z-\dfrac{\rho}{2}|<\dfrac{\...
-1
votes
1answer
24 views

Mclaurin series and n-th derivative

(1) Find the general formula of the McLaurin series of $ f(x) = arctan((x^3)/2)/x^3\ $ (2) Evaluate the 18-th derivative of f(x) (3) Evaluate lim to infinity of f(x) By general formula do we just ...
0
votes
1answer
41 views

Taylor series doesn't seem to have a pattern?

My teacher gave us a study guide to work on, and one of the problems doesn't seem to come out right. The directions are to "find the Taylor series of $f(x)=x^5-3x^4+x^3+2x-1$ for $a=1$. I calculated ...