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. F

    Taylor series (very easy but have a problem)

    Homework Statement series expansion at c=2 of ln(x^2+x-6) Homework Equations The Attempt at a Solution After substituting y= x -2 we get ln(y^2+5y) = ln(y) + ln(y+5) but I am not kinda sure how to use the taylor series of ln(1+x)...
  2. S

    Explanation on Taylor expansions needed

    I have a question about Taylor expanding functions. For both cases I can't get my head around why things are the way they are. I just don't see how one would perform Taylor expansions like that. The first: The starting point of a symmetry operations is the following expansion: f(r+a) = f(r)...
  3. G

    Understanding the Derivation of Taylor Series

    I read wikipedia article also but I can't find the proof of taylor series and from where it came from??
  4. P

    Inverse Laplace transform for small 's', Taylor expansion

    Dear all, This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem: Consider the Laplace transform F(s) = 1 / ( s - K(s) ) , where K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s...
  5. T

    How did Jackson simplify the Taylor expansion for charge density in 3-D?

    On p35 of Jackson's Classical Electrodynamics 3rd Edition, the author gives the expansion of the charge density \rho(\mathbf{x'}) around \mathbf{x'}=\mathbf{x} as \rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ... where r = |\mathbf{x} - \mathbf{x'}| My question is...
  6. S

    Calculating Remainders for Taylor Series of Sine Function

    Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks
  7. S

    Taylor Series for Cosine and Accuracy of Calculating Cosine 2

    Homework Statement How many terms of the taylor series of the cosine function about c = 0 are needed to calculate cosine 2 to an accuracy of 1 / 10000 The Attempt at a Solution I have said that |Rn(2)| = |cosn+1(a) 2n+1/(n+1)!|<2n+1/(n+1!) Now i can't do it ...
  8. J

    A little help with this Taylor expansion please?

    I have this translation operator T(a) that acts on a function y(x) and causes the transformation T(a)y(x) = y(x+a). I am supposed to be "expanding y(x+a) as a taylor series in a" to show that T(a)=eipa, where p is the operator p = -i.d/dx] So, I've started out with the general equation for the...
  9. B

    Understanding Taylor Expansions of Gradients in Vector Calculus

    What does it mean to have a taylor expansion of a gradient (vector) about the position x? I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.
  10. S

    Continuum Conversion of Lattice Points via Taylor Series Expansion

    I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?
  11. M

    What is the correct method for composing Taylor series at a non-zero point?

    in general I'm trying to figure out a way to work with taylor series more efficiently. this means i want to be able to write down the taylor series of a complicated function just by knowing the taylor series(es?) of the component functions. I've figured out how to do products and quotients...
  12. D

    Proving extrema using taylor series and Hessian Matrix

    How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...
  13. Telemachus

    Taylor Development of f(x)=ln(1-x)

    Homework Statement Hi there. I have this exercise which I'm trying to solve now. It says: Using that \displaystyle\sum_{n=0}^{\infty}x^n=(1-x)^{-1} find one Taylor development for the function f(x)=\ln(1-x) So, I've made some derivatives...
  14. I

    Solve Taylor Series Questions with Expert Guidance | Get Help Now!"

    Solved Thanks guys
  15. J

    Using Taylor Series to prove Limit of Exponential Function

    Homework Statement Let v(λ) = ∫0∞ λe-λy f(y)dy a) Prove that limλ→ ∞ v(λ) = f(0). b) Determine and prove limλ→ ∞ λ(v(λ)- f(0)). α β γ δ ε ζ η θ ι κ λ μ ν ξ ο ° π ρ ς σ τ υ φ χ ψ ω Ω ~ ≈ ≠ ≡ ± ≤ ≥ Δ ∇ Σ ∂ ∫ ∏ → ∞ Homework Equations Assume there exists some constant M where maxx∈ℝ...
  16. D

    Taylor series centered at c = 1

    Homework Statement Find the Taylor Series of 1/x centered at c = 1. Homework Equations \sum_{n=0}^{\infty} f^n (c) \frac{(x-c)^n}{n!} The Attempt at a Solution I made a list of the derivatives: f(x) = 1/x f'(x) = -1/x2 f''(x) = 2/x3 f'''(x) = -6/x4 f(1) = 1 f'(1) =...
  17. R

    What is the Taylor series expansion for (1 + x)^.5 and its derivatives?

    Homework Statement a. Find the first four nonzero terms in the Taylor series expansion about x = 0 for f(x) = (1+x)^.5 b. Use the results found in part (a) to find the first four nonzero terms in the Taylor series expansion about x= 0 for g(x) = (1 + x^3)^.5 c. Find the first four...
  18. L

    Finding the Interval of Convergence for the Taylor Series of ln(x) at a=7

    Homework Statement The Taylor series of function f(x)=ln(x) at a=7 is given by: f(x)=\sum^{\infty}_{n=0}c_{n}(x-7)^{n} Determine the interval of convergence The Attempt at a Solution I have worked out that the series would be of the form...
  19. S

    Solving the Taylor Series for e^(-x^2): Is it the Same at x=0?

    Do you just replace the x's with (x-3)'s? Since e^(-x^2) is defined as the taylor series though, it seems like the answer should be the same as the series about x=0. Thanks! P.S. does anyone know how to resize images? :$
  20. V

    Approximating sin(1/2) with Taylor Inequality

    Homework Statement I have to approximate sin(1/2) with the taylor inequality Homework Equations taylors inequality |Rn(x)| ≤ M/(n+1)! | x-a|n+1 The Attempt at a Solution Im not really sure what the significance of this is, but ill do the derivatives f(x) = sin(x) f'(x) = cos(x) f''(x) =...
  21. L

    Coefficients of a Taylor Series

    Homework Statement The function f(x)=ln(10-x) is represented as a power series: \sum^{\infty}_{n=0}a_{n}x^{n} Find the first few coefficients in the power series. Hint: First find the power series for the derivative of . The Attempt at a Solution Okay, start seems fairly...
  22. G

    Finding the limit for exponential function using Taylor Expansions

    Homework Statement Determine the limit and then prove your claim. limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations I know that the formal definition that I need to use to prove the limit is: {limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...
  23. TheFerruccio

    Taylor series for the following

    I have a couple of general questions, combined with this one specific question Homework Statement Find the Taylor or MacLauren series centered about the given value for the following function, determine the radius of convergence Homework Equations \mathrm{Ln}\ z, 2 The Attempt at a Solution...
  24. R

    Taylor Polynomial Approximations

    Homework Statement Let f be a function that has derivatives of all orders for all real numbers. Assume f(1) = 3, f'(1) = -2, f''(1) = 2, and f'''(1) = 4 a. Write the second-degree Taylor polynomial for f about x = 1 and use it to approximate f(0.7) b. Write the third-degree Taylor...
  25. V

    Find the taylor series of ln(1+x)

    Homework Statement find the taylor series of ln(1+x) centered at zero Homework Equations from 0 to infinity ∑ cn(x-a)n cn = f(n)(a)/n! The Attempt at a Solution f(x) = ln(1+x) f'(x) = 1/(1+x) f''(x) = -1/(1+x)2 f'''(x) = 2/(1+x)3 f''''(x) = -6/(1+x)4 f(0) = 0...
  26. M

    Finding Taylor series for x^3 at a = -1

    Homework Statement Find the Taylor series for f(x) centered at the given value of a. (Assume that f has a power series expansion. Do not show that Rn(x)--> 0.) f(x) = x^3, a = -1Homework Equations f(x) = f(a)+f'(a)(x-a)+(f''(a)/2!)(x-a)^2+(f'''(a)/3!)(x-a)^3+...+(f(nth...
  27. E

    How Does Adding a Hydrogen Layer Affect Tunneling Coefficients in STM Analysis?

    Homework Statement In my never ending quest to suck and never be able to do Taylor Expansions, I have another one. I hope one day I'll be able to do these. I have an unknown material and a scanning tunnel microscope. A layer of hydrogen atoms of radius R are added to the surface. This of...
  28. E

    Calculating Circle of Convergence for f(z) = (3z+1)/(15+2z-z^{2}) at z=1

    Homework Statement Expand f(z) = (3z+1)/(15+2z-z^{2}) at z=1 and find the circle of convergence. Homework Equations The Attempt at a Solution I think this is pretty straight forward, but I want to make sure I'm doing everything correctly. I used a power series...
  29. L

    Taylor series analysis question

    Homework Statement [PLAIN]http://img822.imageshack.us/img822/427/scangj.jpg Homework Equations The Attempt at a Solution Hi, could anyone help me with part b of this question, part a I have completed, however I seem to be drawing a blank on the second part
  30. E

    Resources for Addition Taylor Series/Approximations Problems

    First off, I apologize if this is in the spot. I thought about this off and on since this morning on where to put it. There are probably four sections this could go. Feel free to move it if needed. I am awful at recognizing and solving taylor series and approximation type problems. Since...
  31. E

    Taylor Approximation (I think) on Transmission Coefficient

    Homework Statement I have this equation: T=(1+\frac{U_{0}^{2}}{4E(U_{0}-E)}sinh^{2}(2 \alpha L))^{-1} Where α is given by: \alpha = \sqrt{ \frac{2m(U_{0}-E)}{\hbar^{2}}} I have to show that in the limit αL>>1 my equation is approximately given by...
  32. T

    Convergence of a Taylor Series

    Homework Statement Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges? Note: 'sum' represents the sum from n=0 to infinity Homework Equations tanh(z) =...
  33. N

    News Elizabeth Taylor RIP: Movie Legend & Celebrity Icon

    One of the greatest movie legends we've ever had. http://news.yahoo.com/s/ap/us_obit_taylor
  34. L

    Master Taylor Series with Expert Tips | F(E) = E/(KT) + (Ec/E)^1/2

    Homework Statement Expand the function f(E) as a Taylor series.Homework Equations f(E)=E/(KT)+(Ec/E)1/2 The Attempt at a Solution E=Eo So it says that F(E)~Ao+A1(E-Eo)+A2(E-Eo)2... I need to find out what Ao A1 and A2 are, but not sure how to do that. It says as a hint that A1=0 becasue f(E)...
  35. F

    Taylor series for the general distance integral

    Background: I'm trying to transform the gaussian distribution from flat space to curved space. I start with the flat, 1D gaussian distribution in the form \[{\textstyle{1 \over {{{(\pi {\Delta ^2})}^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em...
  36. F

    Understanding Taylor Expansions

    I'm a 2nd year Physics undergrad and am ashamed to admit I don't fully understand the taylor expansion. I have seen the Maclaurin expansion derived which I understand but the Taylor expansion is a little weird to me. I don't really understand what it means to expand a function "about a...
  37. K

    Approximating sin13 with Taylor Series using TI84 (n=150)

    Homework Statement approximate sin13 by using the Taylor series using the TI84. Add for n=150 Homework Equations the infinite sums for sinx is ((-1)^n)(x^(2n+1)/(2n+1)!) The Attempt at a Solution I'm new to programming so i don't have any idea on where to start. I was...
  38. H

    How can Taylor Series help us find a value of h for a specific error tolerance?

    how would you use Taylor Series to answer this: Find a value of h such that for |x|<h implies sin(x)=x-x^3/6 +x^5/120 + R where |R|< 10^-4?
  39. R

    Taylor polynomial of f(x) = 1/(1-x) and the estimate of its remainder

    Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...
  40. L

    Help understanding Taylor expansion etc.

    Well - I am a 16 year old student, whos really interested in math. I do a lot of studying on my own, because I am a bit bored with the present math in school. Right now I am reading about solving differential equations with power series. I can do this, and i do understand the recurrence...
  41. R

    How to Calculate Taylor Approximations for Given Function and Parameters?

    Homework Statement find the 2nd, 3rd, and 6th degree taylor approximation of: f(x) = 10(x/2 -0.25)5 + (x-0.5)3 + 9(x-0.75)2-8(x-0.25)-1 for h = 0.1 to h = 1, with \Deltah = 0.05 and where xo=0; and x = h Homework Equations N.A The Attempt at a Solution I just need to...
  42. N

    How can I simplify this Taylor expansion?

    I want to show this taylor expansion: \frac{1}{\sqrt{1+{x}^{2}}} \rightarrow x^2 what I keep getting is something to the x^3 could some one please help me with this simple expansion?
  43. C

    Taylor polynomial of 1/(2+x-2y)

    Homework Statement Find the Taylor polynomial of degree 3 of \frac{1}{2+x-2y} near (2,1). Homework Equations The Attempt at a Solution I have already solved this problem by evaluating the R^2 Taylor series; I'm mostly curious about another aspect of the problem. By substituting u = x-2y, it...
  44. B

    Finding the First Three Terms of exp(z sin z) Taylor Series

    find the first three non zero terms in the Taylor Series about z=0 of exp(z sin z) i have little idea how to even start on the question because it is exp to the power of z sin z and it just looks too complicated. i hav tried looking thru txtbooks for something similar but no similar question...
  45. H

    I need a method to calculate ln(x) for small x, other than Taylor series method

    I'm aiming to calculate ln(x) numerically. I'm using the following procedure for this: 1) If x is greater or equal to 1, use Newton's method. 2) If x is smaller between 0 and 1, use Taylor series expansion. Newton's method works good, but I have problems with Taylor series expansion method...
  46. F

    Bounding the Error in Taylor Series Approximations for ln(1+x)

    Had a recent homework questions: Find a bound for the error |f(x)-P3(x)| in using P3(x) to approximated f(x) on the interval [-1/2,1/2] where f(x)=ln(1+x) abd P3(x) refers to the third-order Taylor polynomial. I found the Taylor series of f(x) seen below: x- x^2/2!+(2x^3)/3! I know...
  47. Q

    The integral of the taylor expansion of any expression

    ok .lets say the expression we have is ex the taylor expansion becomes 1+x+x2/2+... integrating becomes x+x2/2+x3/6+...+c so how do we know that c = 1? for it to become back to ex becos it is said that integral of ex = ex do we just let x be 0 to find c = 1? does it work for all...
  48. G

    Uses of power series as opposed to taylor series

    So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...
  49. R

    Personal epiphany about Taylor theorem, true?

    Hello, I'm taking my first calculus course right now, and something struck me regarding the remainder in integral form of a Taylor series expansion: Let's say we have a Taylor expansion of the (n-1):th order, which has a remainder of the form Now, my claim is that if we integrate by...
  50. B

    Jackson - p. 35 - taylor expansion of charge density

    Homework Statement With a Taylor series expansion of the well-behaved \rho ({\bf{x'}}) around {\bf{x'}} = {\bf{x}}, one finds the Taylor expansion of the charge density to be, \rho ({\bf{x'}}) = \rho ({\bf{x}}) + {\textstyle{1 \over 6}}{r^2}{\nabla ^2}\rho + ... Homework Equations...
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