Taylor Definition and 878 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. ironman

    Does Taylor Series accurately represent limits in calculus?

    Homework Statement [/B] lim x -> 0 2. Homework Equations Taylor series for sin cos e and ln () The Attempt at a Solution I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this: Now the problem is that WolframAlpha says it should be -6/25. Now if...
  2. RJLiberator

    Evaluating the remainder of a Taylor Series Polynomial

    Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...
  3. V

    How to Apply Taylor Series at Infinity for x >> 1?

    Homework Statement How to use Taylor series for condition x>>1? For example f(x)=x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})} Homework EquationsThe Attempt at a Solution I try to derived it and limit to infinity...for example first term \frac{x^4}{3\sqrt{1+x^2}}. Limit this to infinity is...
  4. RJLiberator

    Basic Taylor Polynomial Question involving e^(-x)^2

    Homework Statement Consider:[/B] F(x) = \int_0^x e^{-x^2} \, dx Find the Taylor polynomial p3(x) for the function F(x) centered at a = 0. Homework Equations Tabulated Taylor polynomial value for standard e^x The Attempt at a Solution [/B] I started out by using the tabulated value for Taylor...
  5. grandpa2390

    How do I use a taylor series expansion to find effective spring constant

    Homework Statement Let's pretend I am given a potential energy function and nothing else. I need to find the effective spring constant for oscillation about the equilibrium point using a taylor series expansion. I can't find an example or explanation anywhere on how to do this. the potential...
  6. P

    Taylor series to find value of nth derivative

    Homework Statement If f(x) = x^5*cos(x^6) find f40(0) and f41(0) The Attempt at a Solution So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?
  7. T

    Taylor Polynomial of 3rd order in 0 to f(x) = sin(arctan (x))

    The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...
  8. I

    Relativistic mechanics Taylor expansion

    Homework Statement For a particle traveling near the speed of light, find the first non-vanishing term in the expansion of the relative difference between the speed of the particle and the speed of light, (c-v)/c, in the limit of very large momentum p>>mc. Hint: Use (mc/p) as a small parameter...
  9. skate_nerd

    MHB Does this problem warrant a taylor expansion? (solid state physics)

    The whole problem I'm doing here is not even really relevant, so I won't go too much into it...I'm told to find an atomic form factor given some certain conditions, and I do a big gross integral and got this: $$f=(\frac{4}{4+(a_oG)^2})^2$$ where \(a_o\) is the Bohr radius and \(G\) is the...
  10. P

    Why Does the k Term Change in Taylor Series?

    Have a quick question about taylor series. We covered taylor series somewhat in class, but there was a complete lack of explanation and our calculus book literally covers the topic in a single page. I understand the idea of a taylor series and how its related to a power series, but what I don't...
  11. T

    Calculating the Taylor Series for cos(x) in Powers of x-pi | Homework Help

    Homework Statement Find the taylor series representation for the following function f(x) = cos(x) in powers of x-pi Homework Equations The Attempt at a Solution [/B] I don't know what they mean by "in powers of x-pi", that's the part I'm confused with. Can somebody please explain that part...
  12. G

    Taylor Series and Random Variables

    Homework Statement A standard procedure for finding an approximate mean and variance of a function of a variable is to use a Taylor Expansion for the function about the mean of the variable. Suppose the variable is y, and that its mean and standard deviation are "u" and "o". f(y) = f(u) +...
  13. C

    Differentiation of a Taylor series

    Homework Statement Hi guys, any help on this question would be hugely appreciated. The Taylor series about 0 for the function f(x)=(1/4+x)-3/2 is f(x)=8 - 48x + 240x^2 - 1120x^3 + ... used differentiation to find the Taylor series about 0 for the function g(x)=(1/4+x)-5/2 The...
  14. K

    Power series absolute convergence/ Taylor polynomial

    1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...
  15. I

    MHB Taylor Series Expansion and Radius of Convergence for $f(x)=x^4-3x^2+1$

    find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?
  16. E

    Another maclaurin vs. taylor series question

    Hey guys, Struggling with understanding this taylor vs. maclaurin series stuff. So a few questions. Let's say that we have some function f(x). 1. By saying that we want to find the power series of f(x) and nothing else, are we implicitly stating that we are looking for a maclaurin...
  17. T

    Calculating Taylor polynomials , Multiple Questions

    Whats up guys ! currently studying for calculas exam and could use someone going over my answers ! Homework Statement Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1) Q.2 Q1...
  18. A

    Cubic approximation multivariable taylor series

    hi everyone , i don't understand these steps for Taylor Expansion , it has used for state space equations the equations are the approximations for sin and cos the equation for Taylor series is ( i don't understand at all ) please help me if you can
  19. B

    Taylor Series Expansion About a Local Minimum

    Hello everyone, I am currently reading chapter two, section 3 of Griffiths Quantum Mechanics textbook. Here is an excerpt that is giving me some difficulty: "Formally, if we expand V(x) in a Taylor series about the minimum: V(x) = V(x_0) + V'(x_0) (x-x_0) + \frac{1}{2} V''(x_0)(x-x_0)^2...
  20. V

    Proving c_{n} = 0 Using Taylor Expansion: Homework Help

    Homework Statement Suppose that f(x)=\sum_{n=0}^{\infty}c_{n}x^{n}for all x. If \sum_{n=0}^{\infty}c_{n}x^{n} = 0, show that c_{n} = 0 for all n. Homework Equations The Attempt at a Solution I know, by using taylor expansion, c_{n}=\frac{f^{n}(0)}{n!}, and because...
  21. M

    Taylor series representation help

    Homework Statement Find the Taylor Series of x^(1/2) at a=1 Homework Equations i have no idea how to do the representation, i believe our professor does not want us to use any binomial coefficients The Attempt at a Solution i got the expansion and here's my attempt at the...
  22. R

    Understanding Taylor Series Approximation with Taylor's Theorem Explanation

    I'm reading a derivation and it says that the following approximation can be used: I do not under stand how Taylor's theorem allows for this approximation. Can anyone explain this a little?
  23. M

    Finding Error on Taylor Polynomials using Taylor's Theorem

    (a) Use Taylor's Theorem to estimate the error in using the Taylor Polynomial of f(x)=sqrt{x} of degree 2 to approximate sqrt{8}. (The answer should be something like 1/2 * 8^{-7/2}. (b) Find a bound on the difference of sin(x) and x- x^{3}/6 + x^{5}/120 for x in [0,1]This is a problem on a...
  24. A

    Where can I learn taylor series and combinatorics?

    I want to learn combinatorics. Please send links? If possible, can you explain now?
  25. B

    Taylor Series coefficient of x^n

    Homework Statement Find the Taylor series for 0.5x^2[e^x-e^(-x)] around x=0. What is the coefficient of x^n? Homework Equations e^x=∑x^n/n! The Attempt at a Solution I understand how to find the Taylor series for this equation (it being ∑[x^(2n+3)/n!]; x^3+x^5+x^7/2!+...) through...
  26. A

    What is the next step for part iii in Taylor Series Extrapolation?

    Hi could anyone give me pointer as to where to go with part iii please?
  27. Conservation

    Difference between Taylor and MacLaurin Series-Usage

    Hi all, I understand the numerical difference between a Taylor and Maclaurin Series; Maclaurin series is just Taylor Series about x=0. However, is there any difference between their usage? I'm guessing Taylor series may be more accurate with less terms for approximating something close to...
  28. Radarithm

    Taylor 4.34 - Energy of a Pendulum

    Edit: Can someone change the name of the thread somehow? I accidentally posted it without changing the name. (Moderator note -- title updated.) Homework Statement The question is quite long so here is a picture: http://gyazo.com/dc917d1885b6ffebb0a39e2409af4d61 Homework Equations...
  29. S

    Help with Taylor Series/Maclaurin Series Question

    Homework Statement Problem is attached in this post. Homework Equations Problem is attached in this post. The Attempt at a Solution I've tried using Maclaurin Series for e^x, and get the term -x^10/5!, however f(0) = 0 which is not the correct answer. Also taking 10 derivatives seems too...
  30. C

    Calculating 13th Taylor Coefficient of f(x)=e^(7x) at x=3

    Homework Statement Determine its 13^{th} Taylor coefficient of the Taylor Series generated by f at x = 3. f(x)=e^(7x) Homework Equations I used the fact that the series for e^x was ∑x^n/n! The Attempt at a Solution Using that above, and replacing x with 7x, shouldn't my answer...
  31. C

    What is the 13th Taylor coefficient of f(x) at x=3?

    Homework Statement F(x)=7x Determine the 13th taylor coefficient of the taylor series generated by f at x=3 Homework Equations Well, it looks like I just had to take the derivative, but by the time it gets to the 13th derivative, wouldn't the answer just be zero? The Attempt at a...
  32. C

    "How do I compute the Taylor series for cos(7x^2) at x=0?

    1. Homework Statement [/b] Determine the Taylor series for the function below at x=0 by computing P 5 (x) f(x)=cos(7x^2) Homework Equations I used to taylor series for cosx and replaced it with 7x^2 so i used 1-49x^4/2! +2401x^8/4!... and so on. That should be correct, my attempt...
  33. A

    Estimating Bandwidth of Phase Modulated Signal Using Taylor Series

    Homework Statement Consider the PM (phase modulated) signal, s(t) = Acos(wt+x(t)) where x(t) is the information bearing signal. Assume that |x(t)|< y, which is not necessarily small. Using Taylor's series expansion, derive an estimate for the bandwidth of the PM signal s(t). Homework...
  34. T

    You can edit the title of the first post and add [solved] at the beginning.

    I'm confused by problem 2.31 in mathematical tools for physics. Problem: 2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler effect for light, including relativistic effects is different still. Show that for low speeds they are...
  35. KiNGGeexD

    Using Taylor expansion to find the limits of a function

    https://www.physicsforums.com/attachments/68247 I had been assigned this problem, I worked out the expansions (for practice) so they could have errors in them! I got to a point (in the photograph) where I could take out a common factor of 1/x but I'm pretty stumped although via other methods...
  36. A

    MHB Taylor Series Applications

    What is the general procedure for using Taylor Series to evaluate: i) sums eg.\sum_{n=4}^{\infty }\frac{n(n-1)2^n}{3^n} ii) limits eg. \lim_{x\rightarrow 2}\frac{x^2-4}{ln(x-1)} iii) derivatives eg. Find f^{(11)}(0) of f(x)=x^3sin(x^2) iv) integrals eg. \int_{0}^{1} \frac{1}{2-x^3}dx
  37. Nugso

    Taylor Approximation: Show ∫f'(x)dx/f(x)=ln|f(x)|+C

    Homework Statement Show that ∫f'(x)dx/f(x) = ln|(f(x)|+C where f(x) is a differential function. Homework Equations First order Taylor approximation? f(x)=f(a)+f'(a)(x-a) The Attempt at a Solution Well, I'm not really sure how to approach the question. It's my Numerical...
  38. F

    Simple Substitution Taylor Polynomials

    I've been taught that with the basic form of a function's maclaurin series, complex forms of the same series can be found. For example, the first three terms for arctan(x) are x-x^3/3 + x^5/5, meaning the first three terms for arctan(x^2+1) at a=0 should be (x^2+1) - ((x^2+1)^3)/3 +...
  39. M

    MHB Linear approximation of Non linear system by Taylor series

    I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...
  40. A

    MHB Taylor Series: Find 2nd Degree Series for x^2+y^2=4 at [1, -\sqrt[]{3}]

    How would I find the second-degree Taylor series for x^2+y^2=4 at [1, -\sqrt[]{3}]?
  41. T

    One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone

    Hi There, I came across the following passage in Sam Glasstone's 'Nuclear Reactor Engineering' See where I underlined in red that taylor series expansion? I don't understand how (dt/dx)_(x+dx) is equal to that. I know it's a Taylor series expansion, but where did the x+dx go?
  42. K

    Derive the analytic expression of a function by its Taylor expansion

    Homework Statement Actually this is not from homework. It occurs in my brain this afternoon. Is it possible to derive the analytic expression of a function by its Taylor series expansion? For example, given the following expansion, how to derive the analytic expression of it? f(x) =...
  43. U

    MHB Manipulating Taylor Expansion for Sample Mean, Variance, Skewness & Kurtosis

    I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...
  44. J

    Struggling with this limit value.(probably using taylor series)

    Struggling with this limit value Homework Statement Calculate lim((e^x-1)/x)^(1/sin(x)) where x\rightarrow0 Homework Equations Maclaurin series. sin(x)/x -----> 1 when x->0 (possibly) The Attempt at a Solution (e^x-1)/x)^(1/sin(x) = ((x+x^2/2+x^3H(x))/x)^(1/sin(x))...
  45. J

    Taylor series in terms of discrete derivative

    All analitic function can be express how: f(x) = \frac{1}{0!} \frac{d^0f}{dx^0}(x_0) (x - x_0)^0 + \frac{1}{1!} \frac{d^1 f}{dx^1}(x_0) (x - x_0)^1 + \frac{1}{2!} \frac{d^2f}{dx^2}(x_0) (x - x_0)^2 + \frac{1}{3!} \frac{d^3f}{dx^3}(x_0) (x - x_0)^3 + ... that is the taylor series of the function...
  46. L

    Taylor series problem (non-direct differentiation?)

    I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?
  47. A

    Clarifications regarding definitions of Taylor, Laurent etc. series

    I want to know the difference between various kinds of series like Taylor, Laurent and Asymptotic. I have some understanding but I want some clarifications. Here is what I understand:- 1) Taylor series is just f(0) + x.f'(0) + x2.f''(0)/2! + ... 2) Laurent series is applied when taylor series...
  48. JasonHathaway

    Taylor series with two variables

    Hi everyone, Homework Statement + Homework Equations + The Attempt at a Solution
  49. M

    Taylor Polynomials and Numerical Analysis

    Homework Statement Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6. *To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use. Homework Equations x = 45 or pi/4, x0 = 42 or 7pi/30 cos(x) = Pn(x) + Rn(x)...
  50. N

    Higher order derivatives with help of Taylor expansion?

    Homework Statement Function f(x) = x^2/(x-1) should be expanded by Taylor method around point x=2 and 17th order derivative at that point should be calculated. Homework Equations Taylor formula: f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)*(x-x0)^2+... The Attempt at a Solution I...
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