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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711 views

Lagrange Error Given a Fourth Derivative

I just would like to check my work with someone else's: The function f has derivatives of all orders for all real numbers, and the fourth derivative of f equals e^(sin(x)). If the third-degree Taylor ...
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2answers
95 views

counting the number of real zeros, and find limitation

Let$$P_n(x)=\sum_{k=0}^{n}(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ let $c_n$ be the number of real zeros of $P_n$. determine$$\lim_{n \rightarrow \infty}\frac{c_n}{2n+1}$$
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61 views

Is it possible to algebraically prove that the $n$th degree Taylor remainder of $f(x)$ is less than $K|\Delta x|^{n+1}$ for $K \in \mathbb{R^+}$?

I found a purely algebraic proof, given below, that for a mononomial $f(x) = x^n$ the magnitude of the error of its linear approximation $| f(x) - [f(a) + f'(a)(x-a)] |$ is less than $K(x-a)^2$ for ...
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1answer
185 views

Orthogonal complete set of functions

Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series). Are there any other orthogonal and complete set of functions for square integrable ...
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1answer
55 views

Exercise about MacLaurin's polynomial and small-o

In class the professor wrote the following limit: $\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$ So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
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3answers
963 views

Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms

Doing a normal Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms by taking derivatives doesn't work because of division by zero. I've put this into wolfram alpha: ...
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1answer
121 views

Differentiating power series

Consider the power series $$\sum_{n=0}^\infty{\frac{x^{2n}}{(2n)!}}$$ From this, it follows that its sum defines an infinitely differentiable function $f$, given by ...
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1answer
62 views

Combining error terms from two Taylor expansions

When deriving the five-point differentiation formula as shown in this book, the IVT was used to combine $ f^{(5)} (\xi_1) $ and $ f^{(5)} (\xi_2) $ into one error term, $ f^{(5)}(\tilde{\xi}) $ As ...
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2answers
567 views

Taylor Polynomial of $\arctan$ of given Degree and Error

Replace the following function by its taylor polynomial of the given grade, and approximate the error in the given interval: $$f(x) = \arctan(x) \textrm{ by } T_3(f,x,0) \textrm{ in } |x| ...
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1answer
88 views

Taylor polynomial of sin of given degree and error

Replace the following function by its taylor polynomial of the given grade, and approximate the error in the given interval: $$f(x) = \sin(x) \textrm{ by } T_3(f,x,0) \textrm{ in } |x| ...
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2answers
872 views

Find the Taylor Series for $f(x)$ centered at a given value $a$

$$f(x) = \frac{6}{x}\,\, \mathrm{at}\,\, a = -4 .$$ Assume that $f$ has a power series expansion. Do not show that $R_n(x) -> 0$ I took the derivatives of f(x): $$f(x) = 6/x$$ $$f'(x) = -6/x^2$$ ...
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1answer
250 views

Base 2 logarithm with Taylor expansion

I'm trying to implement the natural logarithm in C, and our task is to make it really efficient. So what we are doing is, that we use the first 8 members of the series. This works fine, but the ...
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1answer
102 views

Express the function as a Taylor series expansion

How would you find the Taylor series expansion of: $f(x) = \dfrac 3{2x -1} , \text{ at}\, a = 2$
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2answers
139 views

How to show $\prod_{i=1}^n (1-p_i) = \exp{(-\sum_{i=1}^n p_i)} + O(\sum_{i=1}^n p_i^2)$

I have a (seemingly) simple question. How can I see (rigorously) that \begin{equation} \prod_{i=1}^n (1-p_i) = \exp{\left(-\sum_{i=1}^n p_i\right)} + O\left(\sum_{i=1}^n p_i^2\right) \end{equation} ...
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1answer
40 views

Taylor polynomial and degree

I read that one can form Taylor polynomials for some functions, like $$\sin x\approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.$$ Is it correct to say that $\sin x$ has no Taylor polynomial ...
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0answers
290 views

Taylor Series of the Complex Log and Contour Integration

Write the Taylor series of $\text{Log}(1+w)$ with center at $w=0$ on $|w|<1$; check that if $|z-2|<1$, then $|z|>1$. (If you have difficulties in checking this formally, try to draw a ...
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2answers
2k views

Tangent Line Error Bound with Taylor Series

I have an equation, $e^x$, based at 0 (b=0). I am supposed to us the Tangent Line Error Bound to bound the error $|f(x)-T1(x)|$ on the interval I=[-1,1]. (Aside: I have already computed the first ...
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1answer
2k views

taylor expansion of function with a vector as variable

I know how to do a taylor expansion of a function from R to R. I dont know how to do taylor expansion of functions which have 3D vectors as variable. How can I do this? I would appreciate it if ...
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1answer
866 views

What is the nth derivative of $\dfrac{1}{\sqrt{1 + x^2}}$

I'm trying to find a general formula for the $n$th derivative of $$\dfrac{1}{\sqrt{1 + x^2}}$$ I got up to, \begin{eqnarray*} g^{(0)}(x) &=& g(x) \\ g^{(1)}(x) &=& \dfrac{1}{(1 + ...
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1answer
332 views

Electric dipole potential (Taylor expansion)

In the x-y plane, I have a charge of $-e$ at $\mathbf{r} = x \mathbf{i} + y \mathbf{j}$, and another of $+e$ at some point a distance of $\mathbf{s} = s\mathbf{i}$ from $\bf{r}$, such that the ...
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4answers
197 views

Need some information about Taylor Series.

Does a Taylor series always converge to its generating function? Can you please explain? Also, I've encountered an exercise in my Math book. What is the Taylor series generated by a function $ f = ...
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1answer
142 views

Hessian gives a worse approximation of a multivariate function

I have a real, smooth, multivariate (with 10 variables or many more) function, for which I have the exact Jacobian and Hessian. It turns out that unless the norm of the increment of the function is ...
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2answers
141 views

Approximate with error bounds, the integral $ \int^1_0 \dfrac{\sin x}{x}\,dx $

I actually already have the solution to this, but would just like some clarification of how the solution was reached. The solutions provided used the fact that by Taylor's theorem, $\sin x = T_6(x) ...
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1answer
210 views

Approximating arcsin from above

I am very new to function approximations, and I am interested in approximating arcsin with a function $f$, s.t. $f(x) \geq \arcsin(x)$ for all $x$. Taylor series would give me a function which is ...
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0answers
58 views

Error while inverting a function using n terms of its Taylor series

I have $b>a$ and an invertible and infinitely differentiable function $f$. If I want to evaluate $\epsilon=f^{-1}(b-a)$ by writing: $b\approx ...
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235 views

domain of convergence of a multivariable taylor series

consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
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3answers
287 views

difference of square roots approximation

In two of my physics courses in the past week, I've come across an approximation for the difference of two square roots for large radicands: $\sqrt{x+a}-\sqrt{x+b}\approx\frac{a-b}{2\sqrt x}$ for ...
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1answer
300 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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3answers
718 views

Derivation of multivariable Taylor series

I am having trouble grokking why it is, assuming that the function is analytic everywhere (and many other assumptions that I am, no doubt, naively assuming), that this is true: ...
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1answer
91 views

Approximate function from sample data

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. I don't have function definition. It's described as a fuzzy inference system. I have the inference system and can manipulate sample data for each ...
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90 views

About Problem of Taylor series

Carry out Taylor-series deployment of the following function. $$ \begin{align} f(x)&= \operatorname{sec}^2 x \tag{1}\\ g(x)&=\exp \left( -\frac{1}{(1-x)^2} \right) \tag{2} \end{align} $$ I ...
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1answer
296 views

What is the short way of computing the Taylor series using substitution

I know what to do when we have something like x cos(x), you just compute it for cosx and then multiply the terms by x at the end, but what do you do when you have something like ln(3x^2+1)? Do you ...
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0answers
76 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
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12answers
11k views

What are the practical applications of the Taylor Series?

I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: ...
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122 views

Taylor series of a modulus argument

What is the definition of a Taylor series of the function $F(|\vec a -\vec x|)$ about the point $\vec a$ in $\vec x$?
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313 views

Old versus New enunciation of Taylor's Theorem.

I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows: THEOREM Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by ...
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1answer
277 views

How is the error term in the Taylor expansion of a function arrived at?

$f(x)= \sum_{k=0}^{n} \frac{f^k(a)}{k!}(x-a)^k + \frac{ f^{n+1}(c) }{ (n+1)! } (x-a)^{n+1} $ I have some trouble understanding this, as it seems to imply that $ \frac{ f^{n+1}(c) }{ (n+1)! } ...
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141 views

Provide second order taylor expansion of manifolds near equilibrium

I have the following system of equations: $\dot{x}= x + y^2 + yz $ $\dot{y}= -y + z^2 +xz $ $\dot{z}= -2z +x^2 +xy $ I have to provide 2nd order Taylor expansion of 1. the local stable manifold ...
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3answers
128 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
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1answer
94 views

Sequence of polynomials that converges to $|x|$ over $[-1,1]$

Suppose I want to construct a sequence of polynomials that converges to $|x|$ pointwise. I am pretty good on proving that sequences of functions converge to things pointwise, but I am having trouble ...
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2answers
51 views

Can we leave out the first few terms in a Taylor series before doing the ratio test?

http://s12.postimage.org/gljh4ean0/what.jpg we can leave out the first two terms? how come we can do that? is it because the series goes to infinity? can we do that for all taylor series even if we ...
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1answer
150 views

A way of converting non-polynomial form of a function to polynomial form of a function - only taylor expansion?

Taylor expansion allows us to convert non-polynomial form of any function into polynomial form (though the function itself may not be polynomial, as the polynomail can contain infinite terms.). The ...
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1answer
96 views

Taylor expansion of $H = \sqrt{m^2 - \hbar^2 \nabla^2}$

$$ H = \sqrt{m^2 - \hbar^2 \nabla^2} $$ Suppose that there is a equation like this. How do you taylor-expand this equation? I am extremely confused.
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1answer
132 views

Function $k$ times differentiable $+$ root of multiplicity $k$

Problem: Consider the continuous function $f$ which is $k$ times differentiable: $f(\alpha )=f'(\alpha )=\cdots=f^{(k-1)}(\alpha )=0$ and $f^{(k)}(\alpha )\neq 0$. Assume that $\alpha$ is a root to ...
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3answers
893 views

Where do the factorials come from in the taylor series?

Unfortunately, I don't have much detail to give here. But is the general idea to cancel out the constant obtained from taking the derivative. For instance, say my function was ...
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56 views

Question about elementary and nonelementary functions.

Let $E(z)$ be an entire elementary function of (complex) $z$ and $N(z)$ be an entire nonelementary function of (complex) $z$. $e^{N(z)}$$N'(z) = E(z)$ The ' means derivative with respect to $z$. ...
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1answer
92 views

Repeated integration of a real valued bounded function.

I was reviewing the proof the remainder estimate for a Taylor series expansion and I came across something I can't find an intuitive explanation for: if you have a function f that's bounded on an ...
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1answer
831 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
4
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1answer
69 views

How fast does this grow: $f(x) =\sum\limits_{i=1}^{\infty} \dfrac{x^i}{i!^{1/2}}$ for real $x$?

How fast does this grow: $$f(x) =\sum_{i=1}^{\infty} \dfrac{x^i}{i!^{1/2}}, \qquad x\in\mathbb{R}?$$ Is it faster than $e^{x^2}$? If Im not mistaken the Bell numbers grow fast enough such that ...
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2answers
370 views

Why do we need Taylor polynomials?

This question doubles as "Is my understanding of what a Taylor polynomial is for, correct?" but In order to write out a Taylor polynomial for a function, which we will use to approximate said function ...