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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1answer
159 views

Complex Taylor series and Bernoulli numbers

Let: $f(z)=\frac{z}{e^z-1}$ if $z\ne0$, and $f(z)=1$ if $z=0$. Please help to prove that $\sum_{k=0}^{n-1}\binom n kf^{(k)}(0)=0$ for any $n>1$ and $f^{(2n+1)}(0)=0$ for any natrual $n$.
5
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1answer
142 views

Taylor's Theorem Application Question, $f(x)$ smooth and $f(0)=0$ implies $f(x)/x$ smooth.

I am wondering the following fact, and I believe I know the answer, but I am not sure why. If $f(x)$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}$, if $f(0)=0$, is it true that $f(x)/x$ is ...
3
votes
1answer
70 views

What's incorrect with this Taylor series derivation?

Let's \begin{align} f(T)= &f(0)+ \int_0^T f' (t)dt\\ f(T)=&(0)+f' (T)T-\int_0^T f'' (t)tdt\\ f(T)=&(0)+f' (T)t-f'' (T) \frac{T^2}{2}+\int_0^Tf''' (t) \frac{t^2}{2} dt\\ f(T)=&f(0)+f' ...
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votes
1answer
273 views

When finding upper bound for error, can $\xi$ be different from $x$?

The question is to find $P_3(x)$ for $ f(x) = (x-1) \; \ln x $ about $x_0 = 1$ and find the upper bound on the error for $P_3(0.5)$ used to estimate $f(0.5)$. I got $$ f^{(4)}(x)= \frac{2}{x^3} + ...
4
votes
5answers
121 views

$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution?

I recently took an math exam where I had this limit to solve $$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$ and I tought I did it right, since I proceeded like this: 1st I ...
3
votes
2answers
570 views

Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: ...
2
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2answers
66 views

show $\ln \frac{1-x}{1+x}$ is $L^2$ but not $L^1$ using its taylor expansion

I am trying to show that $$ \ln{\Big|\frac{1-x}{1+x}\Big|} $$ belongs to $L^2({\Bbb{R}})$ but not to $L^1({\Bbb{R}})$ by using it's taylor expansion (this is the entire statement of the problem). ...
3
votes
2answers
99 views

Behavior of differential equation as argument goes to zero

I'm trying to solve a coupled set of ODEs, but before attempting the full numerical solution, I would like to get an idea of what the solution looks like around the origin. The equation at hand is: ...
1
vote
1answer
71 views

Taylor theorem for a multivariate BV function

Given a function $f(x_1,x_2,\ldots,x_n):\Omega\to\mathbb{R}$ which is in $BV(\Omega)$ and has, in $\textbf{0}$, all partial derivatives up to order $n-1$, all equal to 0: $$ \frac{\partial^{|\alpha|} ...
6
votes
3answers
394 views

Taylor Series for functions $f:R^n\rightarrow R^n$

I've been told that we can write a taylor series for functions $f:R^n\rightarrow R$ but we can't write one for $f:R^n\rightarrow R^n$. I'm not quite sure why this not possible, but I suspect it have ...
7
votes
1answer
228 views

taylor's formula in $\mathbb{R}^n$ as exponential of derivative operator

I hope this question is not to vague. There is some relationship between the formal operator $$e^\frac{d}{dx} = 1 + \frac{d}{dx} + \frac{1}{2!}\frac{d^2}{dx^2} + \ldots $$ and taylor's formula. Is ...
1
vote
1answer
267 views

Integration on the unit sphere

I have an integral on the unit sphere as follows. $$I(\mathbf{s}_1, \mathbf{s}_2) = \int_{\mathbb{S}^2} f(\mathbf{x} \cdot \mathbf{s}_1)f(\mathbf{x}\cdot\mathbf{s}_2)d\mathbf{x} $$ where the ...
1
vote
3answers
126 views

Taylor expansion of $\sin(x-x^2)$

I am solving a problem that involves expanding $\sin(x-x^2)$. Since $$\sin(x)=x-x^3/6+x^5/120-...$$ i try to substitute $x$ to $x-x^2$ to arrive at expansion of $x-x^2-x^3/6+x^4/2-x^5/2+x^6/6+...$ ...
4
votes
2answers
115 views

$f(y) \leq f(x)+\nabla f(x)\cdot (y-x) $ and $f(x)\geq 0$ implies that $f$ is constant.

Here is the question. Suppose that $f: \mathbb R^n \rightarrow \mathbb R$ has two derivatives and the associated hessian matrix is negative semidefinite on all of $\mathbb R^n$. Show that for any ...
1
vote
1answer
81 views

Integration using Taylor approximations

I am stuck on two problems: 1) Prove that $$\int_0^1 \frac{1+x^{30}}{1+x^{60}}dx=1+\frac{c}{31}$$ where $0< c <1$. 2) Prove that ...
2
votes
2answers
171 views

The Taylor series of $\int_0^x \operatorname{sinc}(t) dt$

I tried to find what is the Taylor series of the function $$\int_0^x \frac{\sin(t)}{t}dt .$$ Any suggestions?
2
votes
3answers
167 views

Inverting $\frac{\log(p-1)}{\log\ p}$

I've been searching for an approximation that will allow me to solve for $p$ in $\frac{\log(p-1)}{\log p}$. Since it is so "obviously" $1 - \frac1{(\text{something})}$ for larger $p$ I thought it ...
2
votes
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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vote
6answers
201 views

How to simplify this expression by division

How to divide it $${\frac {{x}^{n-2}-{y}^{n-2}}{x-y}}$$ to remove the $x-y$ term from the denominator. We may assume that $n>2$ is an integer. Thanks.
2
votes
2answers
62 views

Prove this equation

I'm taking a course on stochastic analysis. I'm stuck on the very first problem of the lecture notes: $\lim_{n \to \infty} \left(1+\frac{\lambda}{n} + o(n^{-1})\right)^n = \exp(\lambda)$ Prior to ...
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votes
2answers
87 views

showing $f$ is entire

Show that the function $f(z)$ is define by $f(0)=1$ and $f(z)=z^{-1}\sin z $ when $z\neq0$, is entire. $\sin z=z-z^3/3!+z^5/5!-z^7/7!...$ (*) We can write $\sin z=z+z^2g_2(z)$ which $g_2(z)$ is ...
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1answer
118 views

Laurent series of $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}=? $$

How to find Laurent series of g(z) ? $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\ 0<a<1 \end{cases} $$ answer is : $$ ...
2
votes
1answer
191 views

How to expand $x \sqrt{4 - x}$ to Maclaurin series?

Here is the task: using standard expansions, expand $f(x) = x \sqrt{4-x}$ to Maclaurin's series. I calculated derivatives up to $f^{(5)}(x)$, and got some results. Fortunately, in Maclaurin's ...
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vote
3answers
75 views

A Taylor Expansion Problem

Show that for all $x>0$ we have $\ln(1+x)>x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}$. I know it has to do with taylor expansion but somehow I cannot prove it rigorously.
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vote
1answer
685 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 ...
4
votes
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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votes
0answers
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 ...
3
votes
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 ...
3
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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 ...
2
votes
3answers
962 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: ...
0
votes
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
565 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| ...
1
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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
864 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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vote
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
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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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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votes
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 ...
2
votes
1answer
860 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 + ...
0
votes
1answer
329 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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vote
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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votes
2answers
140 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) ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
3
votes
0answers
234 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 ...
2
votes
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 ...