# What is Taylor: Definition and 873 Discussions

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. ### Help me with Taylor's theorem please

I am trying to grasp how the last equation is derived. I understand everything, but the only thing problematic is why in the end, it's ##+O(\epsilon)## and not ##-O(\epsilon)##. It will be easier to directly attach the image, so please, see image attached.

5. ### I Can we solve a non-autonomous diffeq via Taylor series?

I've occasionally seen examples where autonomous ODE are solved via a power series. I'm wondering: can you also find a Taylor series solution for a non-autonomous case, like ##y'(t) = f(t)y(t)##?
6. ### Taylor Series Expansion of f(x) at 0

First I got ##f(0)=0##, Then I got ##f'(x)(0)=\frac{\cos x(2+\cosh x)-\sin x\sinh x}{(2+\cosh x)^2}=1/3## But when I tried to got ##f''(x)## and ##f'''(x)##, I felt that's terrible, If there's some easy way to get the anwser?
7. ### Prerequisites for John Taylor classical mechanics

Inside the textbook, the prerequisites state first year mechanics and some differential equations, although it continues to say the differential equations can be learned as you’re working your way through the book, as differential equations were basically “invented” to be used for applied...
8. F

### Link between Z-transform and Taylor series expansion

Hello, I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused. Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
9. ### I Topics covered in John R Taylor Classical mechanics

I can’t find the chapter list online, does anyone know what topics are covered in John Taylor’s classical mechanics? Would it be similar to what’s covered in Newtonian mechanics, but obviously more advanced. Cheers in advance 👍
10. ### B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?

Hi, PF For example, ##\sin{x}=O(x)## as ##x\rightarrow{0}## because ##|\sin{x}|\leq{|x|}## near 0. This fits textbook definition; easy, I think. But, Taylor's Theorem says that if ##f^{(n+1)}(t)## exists on an interval containing ##a## and ##x##, and if ##P_{n}## is the ##n##th-order Taylor...
11. ### Taylor Polynomials question

f(x) = 4 + 5x - 6x^2 + 11x^3 - 19x^4 + x^5 question a almost seems too easy as I end up 'removing' the x^4 and x^5 terms a. T_{2} (x) = 4 + 5x - 6x^2 b. = R_{2} (x) = 11x^3 - 19x^4 + x^5 c. i don't understand what i need to do here. To find the maximum value of a function, we...
12. ### Finding the Taylor series of a function

Greetings! Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm after putting x-1=t we found: But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n Thank you...
13. ### Problem with a taylor serie expansion

Greetings https://www.physicsforums.com/attachments/295843 I really don´t agree with the solution https://www.physicsforums.com/attachments/295846 as I calculated fxy I got fxy=xyexy f(0,1)=0 so x(y-1) should not appear in the solution am I wrong? thank you!
14. ### Multiplication of Taylor and Laurent series

First series \frac{1}{2}\sum^{\infty}_{n=0}\frac{(-1)^n}{n+1}(\frac{1}{p^2})^{n+1}= \frac{1}{2}(\frac{1}{p^2}-\frac{1}{2p^4}+\frac{1}{3p^6}-\frac{1}{4p^8}+...) whereas second one is...
15. ### 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)## $$f_2(s)=(1+s^2)^{1/2}$$...
16. ### I How do I use induction more rigorously when making Taylor expansions?

When I do Taylor expansions, I take the first 3 or 4 derivatives of a function and try to induce a pattern, and then evaluate it at some value a (often 0) to find the coefficients in the polynomial expansion. This is how my textbook does it, and how several other online sources do it as well...
17. ### 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##...
18. ### Bounds of the remainder of a Taylor series

I have found the Taylor series up to 4th derivative: $$f(x)=\frac{1}{2}-\frac{1}{4}(x-1)+\frac{1}{8}(x-1)^2-\frac{1}{16}(x-1)^3+\frac{1}{32}(x-1)^4$$ Using Taylor Inequality: ##a=1, d=2## and ##f^{4} (x)=\frac{24}{(1+x)^5}## I need to find M that satisfies ##|f^4 (x)| \leq M## From ##|x-1|...
19. ### A Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

We sometimes write that \sin x=x+O(x^3) that is correct if \lim_{x \to 0}\frac{\sin x-x}{x^3} is bounded. However is it fine that to write \sin x=x+O(x^2)?
20. ### 4th order Taylor approximation

So I just followed Taylor's formula and got the four derivatives at p = 0 ##f^{(0)}(p) = (1 + \frac {p^2} {m^2c^2})^{\frac 1 2} ## ##f^{(0)}(0) = 1 ## ## f^{(1)}(p) = \frac {p} {m^2c^2}(1 + \frac {p^2} {m^2c^2})^{\frac {-1} 2} ## ## f^{(1)}(0) = 0 ## ## f^{(2)}(p) = \frac {1} {m^2c^2}(1 +...
21. ### 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...
22. ### 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!
23. ### 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...
24. ### I Looking for references on this form of a Taylor series

I was trying to find this form of the Taylor series online: $$\vec f(\vec x+\vec a) = \sum_{n=0}^{\infty}\frac{1}{n!}(\vec a \cdot \nabla)^n\vec f(\vec x)$$ But I can’t find it anywhere. Can someone confirm it’s validity and/or provide any links which mention it? It seems quite powerful to be...
25. ### Evaluate the Taylor series and find the error at a given point

I have the following function $$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$ And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ## I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ## The Taylor polynomial is...
26. ### MHB How to estimate simplex gradient using Taylor series?

I read Iterative methods for optimization by C. Kelley (PDF) and I'm struggling to understand proof of Notes on notation: S is a simplex with vertices x_1 to x_{N+1} (order matters), some edges v_j = x_j - x_1 that make matrix V = (v_1, \dots, v_n) and \sigma_+(S) = \max_j \lVert...

Hi, I was watching a video on the origin of Taylor Series shown at the bottom. Question 1: The following screenshot was taken at 2:06. The following is said between 01:56 - 02:05: Halley gives these two sets of equations for finding nth roots which we can generalize coming up with one...
28. ### I Taylor series and variable substitutions

I'm currently typing up some notes on topics since I have free time right now, and this question popped into my head. Given a problem as follows: Find the first five terms of the Taylor series about some ##x_0## and describe the largest interval containing ##x_0## in which they are analytic...

48. ### How to Evaluate the 8th Derivative of a Taylor Series at x=4

Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
49. ### I What is the tangential component? Taylor p.347

I do not understand why the tan component for a gravity affected by the centrifugal force: g = Ω^2 * R * sinθ * cosθ So I tried to draw this: using a "big" X-shaped axis where the / component goes along the main gravity direction while \ points normal to / this direction. Then the centrifugal...
50. ### Transverse velocity and real/imaginary parts?

Homework Statement The transverse velocity of the particle in Sections 2.5 and 2.7 is contained in (2.77), since By taking the real and imaginary parts, find expressions for v_x and v_y separately. Based on these expressions describe the time dependence of the transverse velocity. Homework...