# 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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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(... 0answers 40 views ### 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} &= ... 1answer 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{\... 1answer 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}}$), but I do ... 1answer 25 views ### Taylor formula with integral remainder in$\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$? Thanks 0answers 63 views ### About Taylor's Theorem in “Calculus 4th Edition” by Michael Spivak I am reading "Calculus 4th Edition" by Michael Spivak. He derived the following proposition on p.423: If$f^{n+1}$is continuous on$[a, x]$, then the following equality holds: $$f(x) = f(a)+... 0answers 18 views ### Bypassing inverse matrix calculation and the comparison of Gradient Descent and Newton Descent I am currently reading Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John. In the 5th chapter the Gradient Descent algorithm is introduced with the following notations : ... 2answers 31 views ### 2 variables Taylor Series expansion at center other than (0,0) When finding the Taylor Series expansion for a function of 2 variables which can be written as a product of two single variable functions, one can multiply their respective Taylor Series expansions to ... 1answer 30 views ### Proof of Maclaurin series of \ln\left(\frac{1}{2-x^2}\right) = \sum_{n=1}^\infty \frac{(-1)^n(1-x^2)^n}{n} I am given the Maclaurin series for \ln(1+x) , namely \ln(1 + x) = \sum\limits_{n=1}^\infty \dfrac{(-1)^{n+1}x^n}{n} , -1 < x \leq 1 . I now need to prove$$ \ln\left(\frac{1}{2-x^2}\... 1answer 36 views ### 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 ... 1answer 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 ... 4answers 156 views ### 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 ... 3answers 57 views ### Maclaurin series of$\frac{1}{(2-x)^2}$When googling about Maclaurin series, I found this: $$\begin{eqnarray*} \frac{1}{1-x} = \sum_{n=0}^\infty x^n \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n \\ \frac{1}{x^2} = \sum_{n=... 1answer 359 views ### Which positive continuous functions satisfy F(x) = F(e^x)-F(e^{-x}) for x\geq 0? There is at least one such function. It is the cdf of the equilibrium probability distribution for the chaotic sequence x(n+1) = |\log x(n)| with x(1) = 2. Its graph (approximation) is pictured ... 0answers 27 views ### Newton Raphson and 2nd Order Taylor [duplicate] In econometrics, while using Newton Raphson algorithm, why do we use 2nd Order Taylor Expansion, what is the requirement or intiution behind it? Is it something related to convergence issue? Can ... 0answers 36 views ### Bounded derivatives implies taylor series (introductory analysis) I've been asked to show that if f is in C^{\infty}, and there exist L>0 such that \vert f^{(n)}(x)\vert \leq L, then for any x_0,x,$$f(x)=\sum_{n=0}^{\infty} f^{n}(x_0)\frac{(x-x_0)^n}{n!... 0answers 33 views ### 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? ... 0answers 52 views ### 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 ... 0answers 39 views ### 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 ... 1answer 27 views ###$_2F_1$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 ... 0answers 36 views ### 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 ... 1answer 22 views ### 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 ... 2answers 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. 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''(... 0answers 23 views ### 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)(... 2answers 80 views ### 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)} + \... 2answers 27 views ### 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-... 0answers 39 views ### 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}},$$where P_k\in \mathbb{Z}\lbrack x\rbrack. It is clear that P_k satisfies the ... 1answer 24 views ### Taylor expansion using Kronecker tensor I have the following function (just used as an example): y_t=g(y_{t-1},\epsilon_t, \sigma) of which I have the following second-order Taylor expansion around a point such that y=\bar{y}, \ \... 3answers 65 views ### Understanding a formula for coefficients a_n of the generating function\sum_{n \ge 0} a_nx^n=\frac{1}{\sqrt{1-x}}. (HMMT 2019 Alge/NT 8) I am trying to understand the solution of this problem, but I don’t understand how the condition described in the title lead to a_n=\frac{_{2n}C_n}{4^n} Does it have anything ... 1answer 30 views ### Taylor Type series I'm stuck in the following series:$$\sum_{n=0}^{+\infty} \frac{1}{n!}\frac{d}{dx^n} \left( f(n-2x) \right) \left|_{x=0} \right.$$where f is a smooth function. At first glance it resembles a Taylor ... 0answers 32 views ### Multivariable Taylor Polynomials and Remainder I searched, but couldnt find any examples. I am given the following function: f(x,y) =sin(sinx+siny) I am required to find the taylor polynomials of order n=1 around a = (0,0), and to ... 1answer 29 views ### Why when finding a Laurent series can we factor the function Say we are trying to find the Laurent series (at zero) of \frac{cos(z)}{z^{3}} why is it we can factor out z^{-3} and multiply the Taylor series expansion of cos(z) at zero when we are trying to ... 1answer 84 views ### Application of Taylor's theorem: find upper bound for remainder? Suppose f is a C^2 function with compact support. I.e. f is 0 outside a closed interval. Then f,f',f'' are uniformly continuous and bounded on \mathbb{R}. My textbook then claims that the ... 0answers 32 views ### Must successive terms in Taylor series expansion become smaller? We use the second derivative test to determine whether a stationary point is max or min. However that is only valid if further expansion of the series does not lead to a reversal in the sign of the ... 0answers 37 views ### How to calculate the Taylor series of the function below for small x? [closed] Could someone help in finding Taylor series of this function$$ \cosh{\sqrt{x^2}}$$? 1answer 41 views ### Truncation error with growing step size When I read about finite difference methods (or really any approximation method), truncation error is often central to the discussion, and rightfully so. But it is also most often discussed in the ... 1answer 26 views ### Find a approximate value of e^(-4) using Taylor series, so that error is less than 10^(-3) So, the Taylor series for e^x is e^x = 1 + x + \frac{x^2}{2!}+ \ldots + \frac{x^n}{n!}. In this instance I have x = -4, so the series looks as follows:$$1 - 4 + \frac{4^2}{2!} - \frac{4^3}{3!} +... 2answers 19 views ###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 ... 1answer 26 views ### 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}}...
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{\...