Questions tagged [taylor-expansion]

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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Singularities of $f(x)=\left(\sqrt{x+1}-\sqrt{x}\right)^{\alpha}$

I'm working with the following function $$f(x)=\left(\sqrt{x+1}-\sqrt{x}\right)^{\alpha}=\left(\frac{1}{\sqrt{x+1}+\sqrt{x}}\right)^{\alpha}$$ with $x\in[0,\infty)$ and $\alpha\in\mathbb{R}$. More ...
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theorem on power series $\log(z^2-iz+2)$

I know how to solve this question.. but dont know using the theorem I know how to find radius of convergence as the wolframalpha says no poles for the log function, how to find then minimum distance ...
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42 views

Taylor polynomial of $f(x_1,…,x_m)=\varphi(e^{a\sum_{i=1}^mx_i})$

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $C^3(\mathbb{R})$ function, with $a\in\mathbb{R}$. Find the Taylor polynomial of degree $3$, centered in the origin $p=(0,...,0)$, of $f(x_1,...,x_m)=\varphi(...
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How do I show that the following solution is stable?

I want to solve the following exercise: Determine the stability properties of the following solution: $$ (\cos t, \sin t) \text{ of }\,\begin{align}\dot{x} &= -y(x^2 + y^2)^{-1/2}\\\dot{y} &= ...
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21 views

Show that a wave signal superposed by a perturbed wave signal can be expressed as combined signal.

Consider the signal where $\delta > 0$ and small, $$y(x,t)=A\sin{(kx - \omega (k) t)} + A\sin{((k + \delta)x - \omega (k + \delta) t)},$$ show that this can be rewritten as $$y(x,t)=2A \cos{\...
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47 views

Taylor Series Expansion in characteristic function

How can we expand the following equation in Taylor series? $\frac{1}{2}$e$^{iu/\sqrt{n}}$ + $\frac{1}{2}$e$^{-iu/\sqrt{n}}$ The solution is 1 $-$ $\frac{u^2}{2n}$ + O($\frac{1}{n^{3/2}}$), but I do ...
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25 views

Taylor formula with integral remainder in $\mathbb{R}^{d}$

Let $V\in \mathcal{C}^{\infty}(\mathbb{R}^{d})$ How to write the Taylor formula with integral remainder of order $n$ for the gradient of $V$ in some element $x_0\in\mathbb{R}^d$? Thanks
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About Taylor's Theorem in “Calculus 4th Edition” by Michael Spivak

I am reading "Calculus 4th Edition" by Michael Spivak. He derived the following proposition on p.423: If $f^{n+1}$ is continuous on $[a, x]$, then the following equality holds: $$f(x) = f(a)+...
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Bypassing inverse matrix calculation and the comparison of Gradient Descent and Newton Descent

I am currently reading Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John. In the 5th chapter the Gradient Descent algorithm is introduced with the following notations : $...
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2 variables Taylor Series expansion at center other than (0,0)

When finding the Taylor Series expansion for a function of 2 variables which can be written as a product of two single variable functions, one can multiply their respective Taylor Series expansions to ...
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Proof of Maclaurin series of $\ln\left(\frac{1}{2-x^2}\right) = \sum_{n=1}^\infty \frac{(-1)^n(1-x^2)^n}{n} $

I am given the Maclaurin series for $ \ln(1+x) $, namely $ \ln(1 + x) = \sum\limits_{n=1}^\infty \dfrac{(-1)^{n+1}x^n}{n} $ , $ -1 < x \leq 1 $. I now need to prove $$ \ln\left(\frac{1}{2-x^2}\...
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Backing out the function knowing its Taylor Series

There are two similar questions in my complex analysis book that I can't for the life of me solve... They are about backing out a function given its Taylor Expansion. I will state both and give my ...
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23 views

Radius of convergence of the given power series.

Let $f$ be a holomorphic function in the open unit disc such that $\lim_{z\to 1}f(z)$ doesn't exist. Let $\sum_{n=0}^\infty a_n z^n$ be the taylor series of $f$ about $z=0$ and $R$ be the radius of ...
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nth derivative of $\frac{e^x−1}{x}$ (both taylor series and finite sum)

Consider the functiong $g(x) =\frac{e^x−1}{x}$. Find a general formula for $g^{(n)}(x)$and prove that this formula is correct. If you want it as a finite sum, Based on guess and check, I think ...
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57 views

Maclaurin series of $\frac{1}{(2-x)^2}$

When googling about Maclaurin series, I found this: $$\begin{eqnarray*} \frac{1}{1-x} = \sum_{n=0}^\infty x^n \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n \\ \frac{1}{x^2} = \sum_{n=...
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Which positive continuous functions satisfy $F(x) = F(e^x)-F(e^{-x})$ for $x\geq 0$?

There is at least one such function. It is the cdf of the equilibrium probability distribution for the chaotic sequence $x(n+1) = |\log x(n)|$ with $x(1) = 2$. Its graph (approximation) is pictured ...
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Newton Raphson and 2nd Order Taylor [duplicate]

In econometrics, while using Newton Raphson algorithm, why do we use 2nd Order Taylor Expansion, what is the requirement or intiution behind it? Is it something related to convergence issue? Can ...
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Bounded derivatives implies taylor series (introductory analysis)

I've been asked to show that if $f$ is in $C^{\infty}$, and there exist $L>0$ such that $\vert f^{(n)}(x)\vert \leq L$, then for any $x_0,x$, $$f(x)=\sum_{n=0}^{\infty} f^{n}(x_0)\frac{(x-x_0)^n}{n!...
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Proof of an equation in Potential theory

I am going through "Foundations of Potential Theory, page 159". In Lemma V, it is stated that: $|\zeta| \leq M (\xi^2+\eta^2)$ Can someone please show the intermediate steps in the proof of Lemma V? ...
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Find Laurent expansion of $\exp(1/z^2)/(z-1)$ around $z=0$ [duplicate]

The question: Find Laurent series of $$f(z)={\exp(1/z^2)\over z-1}$$ around $z=0$. My attampt: $$ \exp(1/z^2)=\sum{z^{-2n}\over n!} \text{ and} \\ {1\over z-1}=\sum z^n $$ I could multiply those ...
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Taylor series of a function $f(x,y)$ about point $(a,b)$

The general expression for Taylor series is given here. My book says we can write the Taylor series about origin $(0,0)$ as: $f(x,y)=\dfrac{1}{2} [f_{xx}\ x^2 + 2f_{xy}\ xy + f_{yy}\ y^2]$ because ...
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$_2F_1$ Expansion around z=1

I want to expand $_2F_1(\frac{1}{2},a,\frac{3}{2},1-x)$ around $x=0$, and then take $a \to 0$. I know that $_2F_1(\frac{1}{2},0,\frac{3}{2},1-x)=1$ for any $x$, Which means only the zeroth value of ...
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Identifying the Hessian Matrix in Taylor's Theorem

In Matrix Differential Calculus with Applications in Statistics and Econometrics, Magnus and Neudecker have a first identification theorem, in which the Jacobian is recognised in the first-order ...
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Taylor series of composition of function with polynomial

Say I have a taylor series around $0$ of some function $f(x) = a_0 + a_1x + a_2x^2 + \cdots$ Say $g(x)$ is a polynomial. Then is it true that taylor series of $h(x) = f(g(x))$ is term by term equal ...
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47 views

General square form of Taylor expansion polynomial

Suppose we have a Taylor expansion for a function $f$ with respect to t up to $M$-th order. $$ \begin{equation} T_M = \sum^M_{k=0}\frac{1}{k!}f^k(x)\Delta t^k = f(x) + f'(x)\Delta t + \frac{1}{2}f''(...
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Undetermined coefficients in a perturbative expansion

In order to familiarize myself with perturbation methods, I've been trying to derive the Lorentz transformations, given by \begin{align*} x \rightarrow \frac{x + vt}{\sqrt{1 - v^2}} & = (x + vt)(...
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$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$

$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$, where $f^{(9)}$ is the $9$-th derivative of $f$. Domain of $f$ is $\mathbb{R} - \{-1, 2\}$. I've got that $f(x) = \frac{1}{1 - (-x)} + \...
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Exponential difference [closed]

I'm solving an exercise of Bernstein theorem, and in one of the passage it goes from a difference between complex exponential to a normal difference. So if O have $e^{it}-e^{is}$, is this equal to $t-...
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Taylor expansion of $1/(1+x^2)^r$ around $x_0$

Let $r>0$, $k\geq 0$. We can write $$\left(\frac{1}{(1+x^2)^r}\right)^{(k)} = \frac{P_k(x)}{(1+x^2)^{r+k}},$$ where $P_k\in \mathbb{Z}\lbrack x\rbrack$. It is clear that $P_k$ satisfies the ...
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1answer
24 views

Taylor expansion using Kronecker tensor

I have the following function (just used as an example): $y_t=g(y_{t-1},\epsilon_t, \sigma)$ of which I have the following second-order Taylor expansion around a point such that $y=\bar{y}, \ \...
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3answers
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Understanding a formula for coefficients $a_n$ of the generating function$\sum_{n \ge 0} a_nx^n=\frac{1}{\sqrt{1-x}}$.

(HMMT 2019 Alge/NT 8) I am trying to understand the solution of this problem, but I don’t understand how the condition described in the title lead to $a_n=\frac{_{2n}C_n}{4^n}$ Does it have anything ...
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Taylor Type series

I'm stuck in the following series: $$\sum_{n=0}^{+\infty} \frac{1}{n!}\frac{d}{dx^n} \left( f(n-2x) \right) \left|_{x=0} \right.$$ where $f$ is a smooth function. At first glance it resembles a Taylor ...
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Multivariable Taylor Polynomials and Remainder

I searched, but couldnt find any examples. I am given the following function: $ f(x,y) =sin(sinx+siny)$ I am required to find the taylor polynomials of order $n=1$ around $ a = (0,0)$, and to ...
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Why when finding a Laurent series can we factor the function

Say we are trying to find the Laurent series (at zero) of $\frac{cos(z)}{z^{3}}$ why is it we can factor out $z^{-3}$ and multiply the Taylor series expansion of $cos(z)$ at zero when we are trying to ...
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84 views

Application of Taylor's theorem: find upper bound for remainder?

Suppose $f$ is a $C^2$ function with compact support. I.e. $f$ is $0$ outside a closed interval. Then $f,f',f''$ are uniformly continuous and bounded on $\mathbb{R}$. My textbook then claims that the ...
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Must successive terms in Taylor series expansion become smaller?

We use the second derivative test to determine whether a stationary point is max or min. However that is only valid if further expansion of the series does not lead to a reversal in the sign of the ...
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How to calculate the Taylor series of the function below for small x? [closed]

Could someone help in finding Taylor series of this function $$ \cosh{\sqrt{x^2}}$$?
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Truncation error with growing step size

When I read about finite difference methods (or really any approximation method), truncation error is often central to the discussion, and rightfully so. But it is also most often discussed in the ...
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Find a approximate value of e^(-4) using Taylor series, so that error is less than 10^(-3)

So, the Taylor series for $e^x$ is $e^x = 1 + x + \frac{x^2}{2!}+ \ldots + \frac{x^n}{n!}$. In this instance I have $x = -4$, so the series looks as follows: $$1 - 4 + \frac{4^2}{2!} - \frac{4^3}{3!} +...
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$U = \frac{h\omega}{e^{hw/KT}-1}\approx KT - \frac{h\omega}{2}+…O(\frac{h\omega}{KT})$

$$U = \frac{h\omega}{e^{hw/KT}-1}\approx KT - \frac{h\omega}{2}+....O(\frac{h\omega}{KT})$$ If have to prove this for $KT\gg h\omega$ I dont understand what the O in the equation means. can someone ...
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the taylor series of the function $f(x) = A/(x-B)^4$ using geometric series.

I have to find the taylor series of the function $f(x) = A/(x-B)^4$ using geometric series. If rewrote it to the general geometric series $\sum x^n=\frac{1}{1-x}$ $A\frac{1}{x-B}=A\frac{-\frac{1}{B}}...
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Best bound for the remainder of two variables Taylor's theorem

Let $f:\mathbb{R}^2 \to\mathbb{R}$ be a two times differentiable function. Then Taylor theorem says that there exists $t\in [0,1]$ such that \begin{align*} f({\bf x})=f({\bf a})+\sum_{i=1}^{2}\frac{\...
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Taylor on $h$ smooth

I'm struggling to proof that $\int_{\mathbb{R}}P(z,t|x)\sum_{n=1}^{\infty}D^{(n)}(z)h^{(n)}(z)dz$ (with $D^{(n)}(z):=\frac{1}{n!}\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_{\mathbb{R}}P(y,\...
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Understanding the variables in Taylor Series

I am looking at the Taylor Series for approximating a function at a given point. Say we have $f(x)$ and $x=0$, then the Taylor Series can be written as... $$f(x) = f(0) + {f}'(0)x + \frac{1}{2!}{f}''(...
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1answer
51 views

Finding value of $K$ when applying the error bound

The problem is asking to use the error bound to find a value of $n$ for which the given inequality is satisfied. $$\left|\sqrt{1.3}-T_{n}(1.3) \right| \leq 10^{-6}, \quad a=1 $$ Now this is how I ...
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24 views

Taylor expansion of fraction involving hyperbolic functions

How can I calculate the Taylor series of this function about $x=0$, $y=0$? $$f(x,y)= \frac{1}{\coth x + \coth y}$$ I can't seem to work out the limits of the derivatives about the origin?
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1answer
70 views

How to solve an indefinite integral using the Taylor series?

I am trying to show that the following integral is convergent but not absolutely. $$\int_0^\infty\frac{\sin x}{x}dx.$$ My attempt: I first obtained the taylor series of $\int_0^x\frac{sin x}{x}...
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1answer
40 views

Question about notation: The order notation

For example, when we Taylor expand say $e^x$ about $x=0$, we would write $$e^x = 1+x+\frac 12 x^2 + \frac 16 x^3 + \mathcal O(x^4) \qquad \qquad \text{as } x \rightarrow 0$$ with the use of the "Big-...
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27 views

Taylor expansion and the intermediate value theorem

Suppose $f : \mathbb R^2 \to \mathbb R$ is a differentiable function at a point $\mathbf a =(a,b) \in \mathbb R^2$. Then for $\mathbf x = (x,y) \in \mathbb R^2$ close to $\mathbf a$, Taylor's theorem ...
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2k views

Taylor series leads to two different functions - why?

Suppose, I want to find a function such that its Taylor series expansion is $$f(x) = \sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+1)a^n}$$ I could start with $$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$ ...