# What is Taylor expansion: Definition + 174 Threads

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

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1. ### Problems with the Riemann tensor in general relativity

After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative. On the internet I found equation (3), and I have tried to understand where this comes from...

The function is $$f(x)=\sqrt{1-x}$$ and we are expected to expand it using Taylor expansion for very small ##(x<<1)## and very big ##(x>>1)## arguments My thought process was the following: $$T_2f[x;x_0]=\sqrt{1-x_0} -\frac 12 \frac 1{\sqrt{1-x_0}}(x-x_0) -\frac 14 \frac 1... 3. ### Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4 let ##k \in\mathbb{N},## Show that there is ##i\in\mathbb{N} ##s.t ##\ \left(1-\frac{1}{k}\right)^{i}-\left(1-\frac{2}{k}\right)^{i}\geq \frac{1}{4} ## I tried to use Bernoulli's inequality and related inequality for the left and right expression but i the expression smaller than 1/4 for any i... 4. ### I Taylor expansion of f(x)=arctan(x) at infinity I have to write taylor expansion of f(x)=arctan(x) around at x=+∞. My first idea was to set z=1/x and in this case z→0 Thus I can expand f(z)= arctan(1/z) near 0 so I obtain 1/z-1/3(z^3) Then I try to reverse the substitution but this is incorrect .I discovered after that... 5. ### I Help with a derivation from a paper (diatomic molecular potential) Hello! I am confused about the derivation in the screenshot below. This is in the context of a diatomic molecular potential, but the question is quite general. Say that the potential describing the interaction between 2 masses, as a function of the radius between them is given by the anharmonic... 6. ### I Taylor Expansion of Metric Transformation in RNCs Carroll expands both sides of metric transformation (Notes eq2.35, Book eq2.48) to equate powers of x’. He starts with eq2.36 (2.49): So far so good, though I feel my understanding of multivariable Taylor series starting to struggle. He refers to Schutz for details, where I find eq 6.23... 7. ### I Why does ##u## need to be small to represent the Taylor expansion Necessary condition for a curve to provide a weak extremum. Let ##x(t)## be the extremum curve. Let ##x=x(t,u) = x(t) + u\eta(t)## be the curve with variation in the neighbourhood of ##(\varepsilon,\varepsilon')##. Let$$I(u) = \int^b_aL(t,x(t,u),\dot{x}(t,u))dt = \int^b_aL(t,x(t) +...

Can you please explain this series f(x+\alpha)=\sum^{\infty}_{n=0}\frac{\alpha^n}{n!}\frac{d^nf}{dx^n} I am confused. Around which point is this Taylor series?
9. ### How to choose the correct function to use for a Taylor expansion?

Consider two different Taylor expansions. First, let ##f_1(s)=(1+s)^{1/2}## $$f_1'(s)=-\frac{1}{2(1+s^{3/2})}$$ Near ##s=0##, we have the first order Taylor expansion $$f_1(s) \approx 1 - \frac{s}{2}$$ Now consider a different choice for ##f(s)## $$f_2(s)=(1+s^2)^{1/2}$$...
10. ### Understanding Taylor Expansion near a Point

I'm just trying to understand how this works, because what I've been looking at online seems to indicate that I evaluate at ##\delta =0## for some reason, but that would make the given equation for the Taylor series wrong since every derivative term is multiplied by some power of ##\delta##...
11. ### Can you use Taylor Series with mathematical objects other than points?

I was recently studying the pressure gradient force, and I found it interesting (though this may be incorrect) that you can use a Taylor expansion to pretend that the value of the internal pressure of the fluid does not matter at all, because the internal pressure forces that are a part of the...
12. ### Problem with series convergence — Taylor expansion of exponential

Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!
13. ### I Taylor expansion of an unknown function

Hello, I have a question regarding the Taylor expansion of an unknown function and I would be tanksful to have your comments on that. Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first...
14. ### I Derive local truncation error for the Improved Euler Method

I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!
15. ### I How to find the moments using the characteristic function?

I have the characteristic function of the Cauchy distribution ##C(t)= e^{-(\mid t \mid)}##. Now, how would I show that the Cauchy distribution has no moments using this? I think you have to show it has no Taylor expansion around the origin. I am not sure how to do this.
16. ### I Struggling with one step to show quantum operator equality

Hello guys, I struggle with one step in a calculation to show a quantum operator equality .It would be nice to get some help from you.The problematic step is red marked.I make a photo of my whiteboard activities.The main problem is the step where two infinite sums pops although I work...

28. ### Charge-Dipole Derivation - Assumption That x >> a

In this derivation: https://cpb-us-e1.wpmucdn.com/sites.northwestern.edu/dist/8/1599/files/2017/06/taylor_series-14rhgdo.pdf they assume in equation (8) that x >> a in order to use the Taylor Expansion because a/x has difficult behavior. Why does that assumption work? Meaning, why can we...
29. ### Derivative of expanded function wrt expanded variable?

Homework Statement If I have the following expansion f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2) This means for other function U(f(r,t)) U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2) Then up to...
30. ### A Questions about the energy of a wave as a Taylor series

I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = ao + a1 A + a2A2 ... Also, that the energy doesn't depend on phase, so only even terms will...

33. ### I Understanding the Taylor Expansion of a Translated Function

I recently found out the rule regarding the Taylor expansion of a translated function: ##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...## But why exactly is this the case? The normal Taylor expansion tells us that ##f(x)=f(a)+f'(a)(x-a)+\frac 1...
34. ### Why do we use Taylor expansion expressing potential energy

My textbook doesn’t go into it, can someone tell me why Taylor expansion is used to express spring potential energy? A lot of the questions I do I think I can just use F=-Kx and relate it to U(x) being F=-Gradiant U(x) but I see most answers using the Taylor expansion instead to get 1/2 kx^2...
35. D

### Taylor expansion fine structure

I have to do a Taylor expansion of the energy levels of Dirac's equation with a coulombian potential in orders of (αZ/n)^2 , but the derivatives I get are just too large, I guess there is another approach maybe? This is the expression of the energy levels And i know it has to end like this:
36. ### A Taylor Expansion of Metric Tensor: Troubles & Logic

Hi, my question is related to taylor expansion of metric tensor, and I have some troubles, I would like to really know that why the RED BOX in my attachment has g_ij (t*x) instead of g_ij(x) ? I really would like to learn the logic...
37. S

### A Taylor expansion of dispersion relation - plasma physics

Hello, I may working through attached paper and really need help with deriving equation in appendix - A4 to give A10. http://iopscience.iop.org/article/10.1088/0004-637X/744/2/182/pdf Any help would be greatly appreciated. thanks, Sinéad
38. ### I Why is There a Dot Product in the Taylor Expansion of 1/Distance with Vectors?

Hi, I would like to express that r and r' are vectors in the attachment and let's say that r is observer distance vector r' is source distance vector. By the way I know this is taylor expansion (for instance if there was only x component (scalar form) I would not any ask question ). But I do...
39. ### B Why Is My Second Derivative Calculation for Taylor Expansion Incorrect?

So in the book it says expend function ƒ in ε to get following. ƒ=√ (1 + (α + βε)2) = √ (1 + α2) + (αβε)/√ (1 + α2) + (β2ε2)/2 (1 + α2)3/2 + O(e3) When I expend I get(keeping ε = 0) ƒ(0) = √ (1 + α2) -->first term ƒ'(0) = (αβ)/√ (1 + α2) --> sec term with gets multiplied by ε for third...
40. ### Recovering the delta function with sin⁡(nx)/x

Homework Statement Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
41. ### Answering "How to Understand Approximation in QM

Homework Statement In the Griffiths book <Introduction to QM>, Section 2.3.2: Analytic method (for The harmonic oscillator), there is an equation (##\xi## is very large) $$h(\xi)\approx C\sum\frac{1}{(j/2)!}\xi^{j}\approx C\sum\frac{1}{(j)!}\xi^{2j}\approx Ce^{\xi^{2}}.$$ How to understand the...
42. ### I Taylor expansions and integration.

I have a short doubt: Let f(x) be a fuction that can't be integrated in an analytical way . Is anything wrong if I expand it in a Taylor' series around a point and use this expansion to get the value of the definite integral of the function around that point? Suppose that the interval between...
43. ### A Taylor/Maclaurin series for piecewise defined function

Consider the function: $$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\ B \exp(-bs) &\text{ if } s>s_c \end{cases}$$ The parameter s_c is chosen such that the function is continuous on [0,Inf). I'm trying to come up with a (unique, not piecewise) Maclaurin series...
44. ### A I need some help with the derivation of fourth order Runge Kutta

Hey guys, I need your help regarding the derivation of the fourth runge kutta scheme. So, I found http://www.ss.ncu.edu.tw/~lyu/lecture_files_en/lyu_NSSP_Notes/Lyu_NSSP_AppendixC.pdf this derivation. Maybe you have a clue what tehy are doing in C.54. So before this they are calculating the...
45. ### Coefficient for a Term in a Taylor Expansion for Cosine

Homework Statement The coefficient of the term (x−π)2 in the Taylor expansion for f(x)=cos(x) about x=π is: Homework Equations ##cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!}...## The Attempt at a Solution Unless my taylor series for cosine is incorrect, I'm...
46. ### A Taylor series expansion of functional

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...
47. ### I Convergence of Taylor series in a point implies analyticity

Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
48. ### First order term in the taylor expansion of ln(x) abut 1

Homework Statement What's the first order term in the expansion ln(x) about x = 1? Homework Equations Taylor series formula The Attempt at a Solution The question is multiple choice, and the choices are x, 2x, or (1/2)x. However, when I calculate the first order term in the expansion of ln(x)...
49. ### Taylor expansion of the relativistic Doppler effect?

[Note from mentor: this thread was originally posted in a non-homework forum, therefore it does not use the homework template.] I have been given an equation for the relativistic doppler effect but I'm struggling to see this as a function and then give a first order Taylor expansion. Any help...
50. ### I Taylor Series: What Is the Significance of the a?

i watched a lot of videos and read a lot on how to choose it, but i what i can't find anywhere is, what's the physical significance of the a, if we were to draw the series, how will the choice of a affect it?