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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3
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2answers
28 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) ...
0
votes
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 ...
0
votes
0answers
22 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 ...
0
votes
3answers
43 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 ...
0
votes
1answer
35 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?
0
votes
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) ...
0
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0answers
28 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 ...
2
votes
1answer
39 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: ...
-1
votes
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?
-3
votes
0answers
25 views

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

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
votes
0answers
84 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 ...
0
votes
0answers
9 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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
29 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
votes
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
votes
1answer
28 views

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

How to find the infinite series expansion of $\arcsin (x)$ and $\arccos (x)$?
0
votes
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 ...
0
votes
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) = ...
3
votes
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 ...
1
vote
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
vote
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 ...
4
votes
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 ...
2
votes
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.
1
vote
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 - ...
6
votes
4answers
162 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
vote
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 ...
0
votes
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' ...
6
votes
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
vote
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 ...
1
vote
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
0answers
61 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
votes
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
30 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 + ...
1
vote
1answer
77 views

What is the limit regarding $a$

What is the limit of : $$ \lim_{x\to 0} \frac{\sin(ax) - \ln(1-2x)}{e^{ax}-1-2x-2x^{2}}$$ I did this with Maclaurin, because my exam is about solving these with MacLaurin. Gave $$\lim_{x\to 0} ...
1
vote
1answer
31 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
1
vote
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} ...
1
vote
1answer
47 views

What can be said about an infinitely differentiable function whose Taylor series diverges?

What can be said in general of an infinitely smooth function whose Taylor series diverges? According to Borel Theorem it is possible to construct such but what kind of property have those special ...
1
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0answers
42 views

Power series and Taylor series

Let $f:\Bbb R\to\Bbb R$ be a $\cal C^\infty$ function. Consider the power series $$\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$$ and call $R$ its radius of convergence. Then, is it true that in ...
0
votes
1answer
79 views

Limiting the error using Taylor's formula with Lagrange's Remainder

I get stuck on problems about approximating values with Taylor's formula when it comes to determining the amount of taylor terms (n) required to satisfy a minumum error by just using Lagrange's ...
0
votes
2answers
28 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
2
votes
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: ...
1
vote
1answer
21 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
1
vote
1answer
23 views

Taylor series of an analytic function that maps the unit disk surjectively onto the upper half plane

Given only that $f(z)$ is analytic and maps the unit disk $|z| < 1$ surjectively to the upper half plane $\Im(z) > 0$, how much can we deduce about $f(z)$? In particular, can we find the radius ...
4
votes
0answers
55 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
1
vote
1answer
37 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 ...