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. F

    Error in Taylor polynomial of e^x

    Find the Taylor polynomial of degree 9 of f(x) = e^x about x=0 and hence approximate the value of e. Estimate the error in the approximation. I have written the taylor polynomial and evaluated for x=1 to give an approximation of e. Its just the error that is confusing me. I have: R_n(x) =...
  2. B

    How Many Terms Needed in Maclaurin Polynomial for Error Below 0.001?

    Taylor Polynomial Error--Please help! Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 So is the procedure to take the derivatives and plug in 0 (since c=0) and find an...
  3. B

    Taylor polynomial approximation- Help

    Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 I really, really don't know what to do for this one, and I have a quiz tomorrow. I have read through the section in the book, but...
  4. B

    Really - Taylor Polynomial Approximation Error

    Homework Statement Use Taylor's theorem to obtain an upper bound of the error of the approximation. Then calculate the exact value of the error. cos(.3) is approximately equal to 1 - (.3)^2/2! + (.3)^4/4! Homework Equations The Attempt at a Solution I came up with upper...
  5. C

    How Do You Calculate a Degree 3 Taylor Polynomial for e^x?

    Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...
  6. A

    Taylor Polynomials: Why Abs(0) Doesn't Have Center at Xo=0

    Why does f(x) = abs (0) not have the first Taylor polynomial center at Xo = 0 ? Does it have second Taylor polynomial center at Xo = 0 ?
  7. O

    Taylor series with 3 variables

    Hi am trying to solve this Taylor series with 3 variables but my result is not equal to the solution- So i think i might be wrong expanding the taylor series, or the solution is not correct Homework Statement Find an a approximated value for the function f(x,y,z) = 2x + ( 1 + y) * sin z at the...
  8. R

    Error Estimate for Taylor Approx of Intergal(f(x)) from 0 to .5

    explain why an estimate of intergal(f(x)) from 0 to .5 using the first 2 nonzero terms of its taylor approximation differs from the actual value of this integral by less than 1/200
  9. B

    Deduce Taylor Series: (2n choose n) x^n Converges to 1/sqrt(1-4x)

    Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...
  10. E

    Taylor Approximation Proof for P(r) using Series Expansion

    [SOLVED] Taylor approximation Homework Statement I have an exact funktion given as: P(r)=1-e^{\frac{-2r}{a}}(1+\frac{2r}{a}+\frac{2r^2}{a^2}) I need to prove, by making a tayler series expansion, that: P(r)\approx \frac{3r^3}{4a^4} When r \prec \prec a The Attempt at a Solution...
  11. E

    Taylor Polynomial of Degree 2 in (0,a): Local Minima Analysis

    Hi, I want to confirm this: a=8 , b=5 , c=7 Decide the Taylor polynomial of degree 2 in the point (0, a) to the function f (x, y)=sqrt(1+bx+cy). Decide with the aid of Taylor polynomial if the function has a local minimum in (0, a). I used the partial derivates: df/dx =...
  12. C

    Applications of Taylor polynomials to Planck's Law

    Due to too much wrong information being posted on my behalf, I am resubmitting a cleaned up version of my last post. I have 2 hours to get this problem done :(. Essentially, I don't know at all how to find the Taylor Polynomial for g(x) = \frac{1}{x^5 ( e^{b/x} -1)} [/URL]
  13. C

    Applications of Taylor polynomials

    Homework Statement f(\lambda) = \frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1} Is Planck's Law where h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s; c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}; and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\...
  14. B

    Upper Bound Error for Maclaurin Polynomial of Sin(x) on the Interval [0,2]

    Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...
  15. O

    Taylor series and quadratic approximation

    Homework Statement use an appropriate local quadratic approximation to approximate the square root of 36.03 Homework Equations not sure The Attempt at a Solution missed a day of class
  16. A

    Compute Taylor Series & Approximate Integral of Exponential Function

    Problem Statement Compute the Taylor Series expansion of f(x) = exp(-x^2) around 0 and use it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt Solution Part1: First to compute the Taylor Series - I am pretty sure about this step so I will not give details...
  17. S

    A few questions about the Taylor series

    When I tried to learn the Taylor series , I could not comprehend why a infinite series can represent a function Would anyone be kind enough to teach me the Taylor series? thank you:smile: PS. I am 18 , having the high school Math knowledge including Calculus
  18. R

    Taylor Series Homework: Find Series for f(x)=sin x at a=pi/2

    Homework Statement Find the Taylor series for f(x) = sin x centered at a = pi / 2 Homework Equations The Attempt at a Solution Taylor series is a new series for me. I believe the first step is to start taking the derivative of the Taylor series. f(x) = sinx f'(x) =...
  19. D

    Taylor Expansion of ln(cos(x))

    \biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +\mathcal{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+\mathcal{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+\mathcal{O}(x^4)\biggr)^3 + \mathcal{O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 +...
  20. T

    Question on solving complex limit with taylor

    question on solving a very hard limit with taylor i want to solve this limit: (1 - (cos x)^(sin(x^2)) ) / (sin (x^4)) i tried to solve it by tailor the problem is that when i subtitute each trigonometric function with a taylor series i get a series in a power of a series i don't...
  21. J

    Taylor series / 2nd deriv test

    Homework Statement Use the Taylor series about x = a to verify the second derivative test for a max or min. Show if f'(a) = 0 then f''(a) > 0 implies a min point at x = a ... Hint for a min point you must show that f(x) > f(a) for all x near enough to a. Homework Equations The Attempt at a...
  22. Y

    Taylor Polynomial Approximation

    How to find a polynomial P(x) of the smallest degree such that sin(x-x^2)=P(x)+o(x) as x->0? Do I have to calculate the first six derivatives of f(x)=sin(x-x^2) to get Taylor polynomial approximation?
  23. J

    What Is the Taylor Series of f(x) and Its Radius of Convergence?

    [SOLVED] power series and taylor Homework Statement Let f be a function defined by f(x)=\frac{1+c x^2}{1+x^2}, and let x be an element of R for c\neq1, find the taylor series around the point a, and find the radius of convergence of the taylor series Homework Equations for power series...
  24. F

    Discovering Maclaurin Series for (1 + x)^(-3) with a Taylor Series Approach

    I am trying to find the maclaurin series for f(x) = (1 + x)^(-3) --> what is the best way of doing this--to make a table and look for a trend in f^(n)?
  25. S

    How Do You Calculate the Taylor Polynomial of Degree 10 for sin(2x) at x=0?

    Find the Taylor polynomial of degree 10 about x=0 for f(x)=sin2x (show all work) This is what i have: M10= f(0)+f\hat{}1(0)x+f\hat{}2(0)x\hat{}2/2!+f\hat{}3(0)x\hat{}3/3!+...+f\hat{}10(0)x\hat{}10/10! f(x)=sin2x f(0)=sin2(0)=0 f\hat{}1(x)=2cos2x f\hat{}1(0)=2cos2(0)=2...
  26. F

    Estimating Accuracy of Taylor Polynomial w/ Taylor Inequality

    We are supposed to use taylor's inequality to estimate the accuracy of the approximation of the taylor polynomial within the interval given. so, f(x) = cos x , a = pi/3, n=4 and the interval is 0<= x <= 2pi/3 the fifth derivative is -sin x to get the M in taylor's inequality, wouldn't...
  27. L

    How Do You Calculate the Maclaurin Series for f(x) = 5(x^2)sin(5x)?

    Homework Statement Find the Maclaurin series of the function f(x) = 5(x^2)sin(5x)Homework Equations \sum(Cn*x^n) The Attempt at a Solution I'm supposed to enter in c3-c7 I already know that c4 and c6 are 0 because the derivative is something*sin(0)=0 but for the odd numbered c's I am having...
  28. B

    Taylor Series Expansion of g(z)=1/(z^3) About z0=2

    Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...
  29. F

    Taylor polynomial approximation (HELP ME)

    Ok, we are asked to determined the degree of the the taylor polynomial about c =1 that should be used to approximate ln (1.2) so the error is less than .001 the book goes throught the steps and arrives at: |Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1) but then, it states that...
  30. C

    Discover the Taylor Series for 3/(z-4i) about -5 | SOLVED

    [SOLVED] Taylor Series Question I have to find the Taylor series of \frac{3}{z-4i} about -5. Therefore, we want the series in powers of z+5. Now, following the textbook it appears that we want to get this in a form that resembles a geometric series so that we can easily express the Taylor...
  31. M

    How Does Taylor's Series Apply to Multivariable Functions?

    Homework Statement http://img99.imageshack.us/img99/9044/tayloriq0.th.jpg Homework Equations ? The Attempt at a Solution I have no idea - please help...
  32. M

    Taylor Series for sinx about pi/6

    Homework Statement Determine the Taylor Series for f(x)=sinx about the center point c=pi/6Homework Equations pn(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ...The Attempt at a Solution f(pi/6) = 1/2 f'(pi/6) = \sqrt{3}/2 f''(pi/6) = -1/2 f'''(pi/6) = -\sqrt{3}/2 f(4)(pi/6)...
  33. F

    What are they asking about this Taylor series?

    I'm unclear on what they are asking in this homework problem. Suppose we know a function f(z) is analytic in the finite z plane apart from singularities at z = i and z=-1. Moreover, let f(z) be given by the Taylor series: f(z)=\displaystyle\sum_{j=0}^{\infty}a_{j}z^{j} where aj is...
  34. C

    How to Find the Bloch Vector for a Density Matrix Using Taylor Expansion?

    Homework Statement I need to find the bloch vector for the density matrix \frac{1}{N}\exp{-\frac{H}{-k_bT}} where the Hamiltonian is given by H=\hbar\omega\sigma_z. The Attempt at a Solution I can break the Taylor series of exp into odd and even terms because sigma z squared is the...
  35. R

    Taylor Series Approximation Help

    Homework Statement Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem: y'= cosh(4x^2-2y^2) y(0)=14 And write a computer program to do the grunt work approximation Homework Equations The Attempt...
  36. R

    Proving Taylor Series: Maclaurin vs. Taylor

    How does one prove taylor series? Is it proven the same way as Maclaurin's Series(Which i know is a special case of taylor series) f(x)=A_0+A_1x+A_2x^2+A_3x^3+... f(\alpha)=A_0+A_1\alpha+A_2(\alpha)^2+A_3(\alpha)^3+... this kinda doesn't seem like a good way to prove it...as that is how I...
  37. I

    How to Find the Taylor Polynomial of a Function Composition?

    Is there any nice trick for finding the Taylor polynomial of a composition of 2 functions, both of which can be expressed as taylor polynomials themselves? For example, finding the taylor polynomial for e^{\cos x}. Thanks.
  38. N

    Show Taylor Formula Proves E > T_2_E for 0 to c

    Homework Statement I have E(v) = (m*c^2)/sqrt(1-v^2/c^2). I also have a second-order Taylor-polynomial around v = 0, T_2_E, which is mc^2+½mv^2. I have to use Taylors formula with restterm to show that E is bigger than T_2_E for all v in the interval [0,c). The Attempt at a Solution...
  39. N

    Why Must the Second-Order Taylor Polynomial for E(v) at v=0 Equal mc^2 + ½mv^2?

    I have E(v) = (mc^2)/(sqrt(1-(v^2/c^2)). I have found the second-order Taylor-polynomial for v=0, and I get: T_2_E(v) = mc^2 + ½mv^2. My teacher asks me, why this equation must be true - what is so special about the second order Taylor-polynomial for v = 0 for E(v)?
  40. P

    1st degree taylor polynomial question

    Homework Statement find an interval I such that the tangent line error bound is always less than or equal to 0.01 on I f(x) = ln(x) b = 1 The Attempt at a Solution so basically, i found the tangent line approximation at b = 1, which is t(x) = x -1. From there though, i have no idea...
  41. P

    Very very basic taylor series problem

    Homework Statement Consider f(x) = 1 + x + 2x^2+3x^3. Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function Homework Equations The Attempt at a Solution This is the first time that I have seen Taylor series and I am totally lost on how to...
  42. F

    Evaluate the limit of.What is O(x^2) and how can it be used to evaluate a limit?

    Homework Statement I've been asked to: Use the real Taylor series formulae e^{x} = 1 + x + O(x^{2}) cos x = 1 + O(x^{2}) sin x = x(1 + O(x^{2})) where O(x^{2}) means we are omitting terms proportional to power x^{2} (i.e., \lim_{x\rightarrow0} \frac{O(x^{2})}{x^{2}} = C where C is a...
  43. C

    Deriving Taylor Series: Understanding the Step Escaping Me

    I was going through the derivation of the Taylors series in my book (Engineering Mathematics by Jaggi & Mathur), and there was one step that escaped me. They proved that the derivative of f(x+h) is the same wrt h and wrt (x+h). If someone could explain that, Id be really grateful.
  44. B

    Why Is a Taylor Series More Accurate Near Its Expansion Point?

    Hi I have some questions. If you're doing a MacLaurin expansion on a function say sinx or whatever, if you take an infinite number of terms in your series will it be 100% accurate? So will the MacLaurin series then be perfectly equal to the thing you're expanding? Also I don't really...
  45. S

    How Do You Determine the nth Term of a Taylor Polynomial for ln(1-x)?

    Homework Statement Write down the Taylor Polynomial of degree n of the function f(x) at x=0 Homework Equations f(x) = ln(1-x) The Attempt at a Solution f(x) = ln(1-x) f'(x) = (-1)((1-x)^(-1)) f``(x) = (-1)((1-x)^(-2)) f```(x) = (-2)((1-x)^(-3)) f````(x) =...
  46. Schrodinger's Dog

    Is the Taylor polynomial for ln(1-x) the same as the one for ln(1+x)?

    Homework Statement ln(1+x)=x-\frac{1}{2}x^2+ \frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5-... -1<\ x\ <1 Is there a Taylor polynomial for ln(1-x) for -1< x <1, if so how would I go about working it out from the above? This is not really a homework question just a thought I had, as they do it...
  47. E

    Understanding Big O Notation in Taylor Series

    Homework Statement Can someone explain big O notation to me in the context of taylor series? For instance, how do you know that sint t = t - t^3/(3t)! + O(t^5) as t -> 0? Does that hold when t -> infinity as well? Is there a generalization of this rule? Is it derived from the...
  48. W

    Geometrical interpretation of Taylor series for sine and cosine?

    I've stumbled upon what might be a geometrical interpretation of Taylor's series for sine and cosine. Instead of deriving the Taylor's series by summing infinite derivatives over factorials, I can derive the same approximation from purely geometrical constructs. I'm wondering if something...
  49. J

    How to Find a Taylor Series for an Integral with x=-1 as Center Point?

    Homework Statement Hi everyone, determine a Taylor Series about x=-1 for the integral of: [sin(x+1)]/(x^2+2x+1).dx Homework Equations As far as I know the only relevant equation is the Taylor Series expansion formula. I've just started to tackle Taylor Series questions and I've been...
  50. L

    The Taylor series expansion for sin about z_0 = (pi/2)

    Homework Statement Expand cos z into a Taylor series about the point z_0 = (pi)/2 With the aid of the identity cos(z) = -sin(z - pi/2) Homework Equations Taylor series expansion for sin sinu = \sum^{infty}_{n=0} (-1)^n * \frac{u^{2n+1}}{(2n+1)!} and the identity as given...
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