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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1answer
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error order evaluation in taylor expansion of a definite integral

I have a function $g(x)=f(x)e^{-x}$ and i want to consider the following integral: $\int_{0}^{\infty}g(x)dx$. Since $f(x)$ is a complicated, but monotonic decreasing, function in the interval ...
0
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1answer
22 views

Prove error bound using Taylor's series Error term (Bound doesn't seem to make sense)

I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is ...
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1answer
25 views

I do not know the point at which this Taylor series was derived, can someone explain please?

I am required to derive Euler's method through Taylor's Theorem. I have been given the Taylor series for $y(t)$ as shown below. However I do not understand what point the Taylor series was derived. ...
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1answer
12 views

Error estimate of definite integral of a taylor expanded function

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute ...
2
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1answer
44 views

Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
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5answers
3k views

Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
3
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2answers
29 views

Other ways to evaluate $\lim_{x \to 0} \frac 1x \left [ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1\right ]$?

Using the facts that: $$\begin{align} \sqrt{1 + x} &= 1 + x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt{1 - x} &= 1 - x/2 - x^2/8 + \mathcal{o}(x^2)\\ \sqrt[3]{1 + x} &= 1 + x/3 + \mathcal{o}(x) ...
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3answers
45 views

(Taylor's theorem) Proving that $\sin(x) = \sum\limits_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}$

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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1answer
19 views

approximation of x=sin(x) error

The larger $x$ becomes, the worst the approximation of $x=\sin(x)$ becomes. How large can $x$ get before the error is no longer less than $1/(2\times10^{14})$? What I have tried: I know that the ...
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0answers
23 views

Finding a limit using Taylor's theorem

let's say that g(x,y) is $c^{n+1}$ and let's say that p(x,y) is it's n-th order Taylor polynomial. I am trying to prove that: $$\lim_{(x,y)\to (0,0)} \frac{g(x,y)-p(x,y)}{(\sqrt{x^2+y^2})^n}=0$$ I ...
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1answer
36 views

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$.

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$. What is its radius of convergence? So I write the fraction as $\frac{1}{(x-1)(x+3)}$ and what should I do now?
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0answers
31 views

Taylor Polynomial Accuracy [on hold]

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say x=0). So I'm considering the function 1/(1-x) centered at 0. I have found ...
0
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0answers
12 views

cos(x) approximation with taylor of second degree

there is an approximation to find cos(x) is 1 - (x^2)/2, until n = 2 degree of taylor, but I'm confuse how to find how good is its approximation, the one thing I know only I get its error is (sin(c) ...
4
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6answers
454 views

Taylor expansion of $e^{\cos x}$

I have to find the 5th order Taylor expansion of $e ^{\cos x}$. I know how to do it by computing the derivatives of the function, but the 5th derivative is about a mile long, so I was wondering if ...
-1
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1answer
53 views

How to derive Maclaurin series for ln(1+x) without calculus?

How can we show $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$ for $-1 < x \leq 1$ without using calculus?
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1answer
82 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
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0answers
26 views

the question is Consider the function f(x)= (1-cos x)/x [closed]

1) what is the correct definition of f(0): that is the value makes f continuous? 2) near what points is there a loss of significance if the given formula is used? ...
6
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0answers
86 views

Taylor expansion of $x^{1/x}$

I am new to Taylor expansions and I would like to calculate the Taylor polynomial of the function $x^{1/x}=e^{(1/x)\log x}$. Since the function is not defined at $x=0$, how should I choose the point ...
2
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3answers
60 views

Does the Taylor series of an infinitely differentiable function converge; and if yes, does it converge to the function? [duplicate]

I have googled it, but I am not satisfied with those. So my questions are: Let $D$ be an open set in $\mathbb{R}$. Let $f:D\rightarrow \mathbb{R}$ be a infinitely differentiable ...
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0answers
10 views

Expansion at Infinity of the Error Function

Since I am workin with some integrals involving Brownian motion I need to compute the expansion of $$ f\left(\Delta\right)=\textrm{erf}\left(\frac{d}{\sqrt{\sigma_0^2\,\Delta}}\right) $$ when ...
0
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1answer
13 views

Convergence of Taylor series about centre of open disc for analytic function.

I define a function on an open set of the complex plane to be analytic if about any point $z_0$ in that set it can be expanded as a power series in $(z - z_0)$ that converges in some neighbourhood of ...
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0answers
5 views

Taylor Polynomial to estimate solution of MVC differential equation

I have read a few examples, that you're using derivatives at different points to estimate a polynomial but I need a bit a of guidance to understand how this would work in a multivariate calculus ...
4
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1answer
375 views

Taylor series for cosine around $\pi/3$

I need the Taylor-Series for $ f(x) = \cos(x) $ in $ a = \pi/3$: \begin{align*} f(x) &= \cos(x - \pi/3 + \pi/3) \\ &= \cos \left( x - \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) - ...
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1answer
30 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...
0
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1answer
53 views

Sum of Taylor Series

I have the converging series: $$ 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... $$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
0
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1answer
29 views

Infinite series expansion of $\arcsin (x)$ and $\arccos (x)$

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?
2
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4answers
148 views

How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large?

How can the values of the function $f(x) = \sqrt{x + 2} −\sqrt{x}$ be computed accurately when $x$ is large? I have tried using Matllab. I am not able to understand when $x$ will be large.
4
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3answers
171 views

How is the formal inverse of a power series with constant term developed ( for instance $\cosh^{-1}(x)$)?

In an older question here in MSE I've asked for the term for the "slicing" of a power series in partial series and have learned that it is "multisection". I' ve been looking at the behaviour of the ...
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0answers
21 views

Taylor Series Expansion

PROBLEMS ax^2 + x + 1 = 0 (1) 1. Using a Taylor series expansion express the solution to the quadratic equation in Equation (1) as a series. Include terms up to cubic order. Find the cubic term in ...
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0answers
15 views

Is there an expression for $\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = $?

Does an expression for $$ \exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ? $$ exist? For j=1 we have the usual expression for translation and scaling $$ \exp\left( t \partial_z\right) f(z) = ...
1
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1answer
33 views

Series representation of function with fractions, logarithms, squares and cosines.

I'm looking for a series representation for $$\dfrac x{x^2+(\log \cos x)^2}$$ Where $x\in(0,\pi/2)$ Note: Both finite and infinite series are accepted. I have tried taylor series, but it requires ...
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4answers
48 views

Taylor polynomial of $\frac{1}{2-x}$

Can someone show how to find the Taylor polynomial of $\frac{1}{2-x}$? I tried this: $\frac{1}{2-x}=\frac{1}{1-(x-1)}$ and then use that $ \ T_n(\frac{1}{1-x})=1+x+\dots +x^n.$ But this gives ...
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0answers
26 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
1
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2answers
287 views

Determine whether a multi-variable limit exists $\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$

I need to determine whether the next limit exists: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$ Looking at the numerator $(-1-\frac{x^2}2)$ it immediately ...
6
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4answers
165 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
1
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1answer
15 views

Estimating the remainder for Mac Laurin's series

I'm practicing Taylor's series and i found some old task. Calculate value of function $f(x) = e^x + e^{-x}$ at point $x = \frac{1}{\sqrt 2}$ with error not greater than $d=\frac{1}{20}$ So here's ...
6
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3answers
133 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
1
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1answer
50 views

Differentiability of the remainder in Taylor's theorem

Suppose we have a function that's differentiable $m$ times over $[a,b]$, we have $a< \alpha < x < b$ and $n < m$. Then $$ f(x) = \sum_{i = 0}^{n-1} \frac{f^{(i)}(\alpha)}{i!}(x - ...
5
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2answers
169 views

Multiple differentiability from Taylor expansion

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a real function, and let $0\leq n\leq+\infty$. We make the following assumption: For every $a \in\mathbb{R}$ and for $k=n$ (resp., in the case $n=+\infty$: ...
0
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1answer
31 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
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0answers
22 views

Taylor's remainder in a compact

My impression is that each function enough regular ($C^\infty$ ) in a compact is equivalent to a polynomial. Is this true? Is there a way to prove it? The expression of the Taylor's remainder just ...
2
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1answer
35 views

taylor series expansion, derivatives not continuous

As a part of an excercise I am supposed to find the Taylor series expansion for $(1-t)^{\frac{1}{2}}$ on $[0,1]$. According to the remainder theorem: ...
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2answers
15 views

Gradient decent using Taylor Series

I'm reading a book about Gradient methods right now, where the author is using a Taylor series to explain/derive an equation. $$ \mathbf x_a = \mathbf x - \alpha \mathbf{ \nabla f } (\mathbf x ) $$ ...
2
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1answer
46 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
2
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1answer
45 views

$f$ differentiable and $f(0)=f(1)=0$. , prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

Let $f$ be differentiable on $[0,1]$ and $f(0)=f(1)=0$. Also, we know $|f''(x)| \le A$ on $(0,1)$, prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$ I'm guessing I should use taylor ...
3
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1answer
2k views

First four Taylor Series Expansions

I'm supposed to write the first four taylor series expansions of $f(x=0)$ using: one term, two terms, three terms, four terms This is the function: $$f(x) = x^3 - 2x^2 + 2x - 3$$ Should I be using ...
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0answers
28 views

What is the bound on the error that is given by Taylor's inequality?

I have this problem here that I'm not sure how to solve. Consider the polynomial $1 - \large\frac {x^2}{2!}$ as an approximation to $\cos(x)$ on the closed interval $-1 \le x \le 1$. What is the best ...
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0answers
63 views

Taylor series expansion and Laplace transform final value theorem

I cant figure out how some transformations are made in one article on physics. Here is expression in s-domain and they want to find its asymptotic value. $$ \xi(s) = \nu_1(s+1)=\frac{1}{(s+1)} ...
2
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1answer
32 views

Evaluating $\ln(\cos x))$ using Taylor expansion

Evaluate $\ln(\cos x)$ at $x_0=0$ and with the order of $n=4$. Noticing that $\ln(\cos x) = \ln(1+ \cos x - 1)$ we can use $\ln(1+x)$ Taylor series. Now, I've read I should use: $$\ln(1+x) = x - ...
2
votes
1answer
31 views

Using Taylor series with remainder

Arfken and Weber expand $(1+n^{-2})^{-1}$ as $$(1+n^{-2})^{-1}=1-n^{-2}+n^{-4}-\frac{n^{-6}}{1+n^{-2}}$$ However, if I use Taylor/Maclaurin expansion $$(1+x)^m=1+mx+ \frac{m(m-1)}{2!} x^2 + \cdots + ...