What is Taylor series: Definition and 492 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. G

    Laurent and Taylor series in the unit disc

    Homework Statement Let f(z) be a function that is analytic for all |z|≤1, with the exception of z_0, which lies on the circle |z|=1. f(z) has a first order pole at z_0. Letting Ʃ a_n z^n be the Maclaurin expansion of the function, prove that z_0 = lim_(n→∞) a_n/a_(n+1) Homework Equations...
  2. T

    Taylor Series Expansion to Compute Derivatives

    Homework Statement Find the Taylor series expansion of f(x) = (x-1)/(1+(x-1)^2) about x=1 and use this to compute f(9)(1) and f(10)(1) Homework Equations The sum from n=0 to infinity of f(k)(c)/(k!) (x-c)k The Attempt at a Solution I'm not sure how to approach this...
  3. D

    Hyperbolic sine in Taylor Series

    I am reading through a worked example of the Taylor series expansion of Sinh(z) about z=j*Pi The example states: sinh(j*Pi)=cos(Pi)*Sinh(0) +jcosh(x)sin(y) I am unsure of this relation. I understand why the x terms are zero but don't know the relation to expand sinh. Can anyone shed...
  4. H

    A question about Taylor series expansion

    Homework Statement Find the Taylor series expansion for f(x)=x*e^(-x^2) about x = -1 Homework Equations The Attempt at a Solution I have tried replacing x with (x-1) and f(x-1) = (x-1)*e^(-(x-1)^2). Consider the power series for e^(-(x-1)^2) about x = 0, f(x-1) =...
  5. H

    A question about Taylor series expansions

    Find the Taylor series expansions for f(x)=x*e^(-x^2) about x = -1 -(1/E) - (x + 1)/E + (x + 1)^2/E + (5 (x + 1)^3)/(3 E) + (x + 1)^4/( 6 E) - (23 (x + 1)^5)/(30 E) - (29 (x + 1)^6)/(90 E) + ( 103 (x + 1)^7)/(630 E)... This is the answer from Mathematica but i don't know how it goes. Can...
  6. L

    Summing a series- Taylor series/ complex no.s?

    Homework Statement Find the sums of the following series: S1=1+(x^3)/(3!)+(x^6)/(6!)+... S2=x+(x^4)/(4!)+(x^7)/(7!)+... S3=(x^2)/(2!)+(x^5)/(5!)+(x^8)/(8!)+... Homework Equations Perhaps Taylor series? The Attempt at a Solution I spotted that adding S1+S2+S3=e^x, but I don't...
  7. A

    Taylor Series Expansion for f(z) = −1/z^2 about z = i + 1

    Homework Statement Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1. Homework Equations The Attempt at a Solution I'm just not sure what format I'm supposed to leave it in. Is it meant too look like this: f(z)=f(i+1)+f'(i+1)(x-i-1)... or this Ʃ\frac{1}{n!}f^{(n)}(1+i) *...
  8. N

    Taylor series and second derivative test: the degenerate case.

    Hello! I am wondering if someone could let me know if my understanding is right or wrong. The Taylor series gives the function in the form of a sum of an infinite series. From this an approximation of the change in the function can be derived: f_{a} and f_{a,a} are the first and second...
  9. A

    What is the trick to simplifying the Taylor series of 1/(1 + x^2)?

    The equation starts at B and this is my attempt. As you can see it soon complicates and doesn't look like what t should since I already know what the Taylor series of his function should look like. Is there some clever trick to it that I am missing? PS the series is centred around x = 0...
  10. T

    Taylor series at a point for which the function isn't defined (perturbation)

    Homework Statement This problem arises from the following ODE: \epsilon y'' + y' + y = 0, y(0) = \alpha, y(1) = \beta where 0 < x < 1, 0 < \epsilon \ll 1 Find the exact solution and expand it in a Taylor series for small \epsilon Homework Equations I guess knowing the Taylor...
  11. O

    Taylor Series: Show Terms Decay as 1/n^2

    Show that, with an appropriate choice of constant c, the taylor series of (1+cx)ln(1+x) has terms which decay as 1/n^2 I know that ln(1+x) decays as 1/n, but I don't know how to show the above. Please help. Thanks in advance
  12. D

    How can the Taylor series help prove the limit of cosine?

    I have to prove that \cos(x) = 1 - \frac{x^2}{2} + O(x^4) (x \to 0) My ugly attempt: \lim_{x \to 0} \frac{\cos(x) - 1 + \frac{x^2}{2}}{x^4} \lim_{x \to 0} \frac{\cos(x) - 1}{x^4} + \frac{1}{2x^2} \lim_{x \to 0} \frac{\sin(x)}{4x^3} + \frac{1}{2x^2} \lim_{x \to 0}...
  13. J

    Finding Taylor Series - different Method

    Homework Statement Hello, I'm in the middle of solving for the Taylor series of the function: f(x)=sin(2x)ln(1-x) up to n = 4. The Attempt at a Solution So far, I've been strictly taking its derivatives until I reach the fourth. It's becoming a very long process considering it's...
  14. N

    Complex analysis, taylor series, radius of convergence

    Homework Statement For f(z) = 1/(1+z^2) a) find the taylor series centred at the origin and the radius of convergence. b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius. Homework...
  15. R

    Taylor series and the forward finite difference method

    Given a partial differential equation, how would one go about implementing the forward finite difference method to the Taylor series?
  16. C

    Taylor Series : How to determine coefficient

    Homework Statement https://skydrive.live.com/?cid=6b041751c72e14ad#!/?cid=6b041751c72e14ad&sc=photos&uc=3&id=6B041751C72E14AD%21149!cid=6B041751C72E14AD&id=6B041751C72E14AD%21154&sc=photos The Attempt at a Solution...
  17. R

    Using Taylor Series for Initial Value Problems

    Homework Statement I posted this already but decided to revive this thread since I re-worked the problem. Consider dy/dx=x+y, a function of both x and y subject to initial condition, y(x0)=y0. Use Taylor series to determine y(x0+\Deltax) to 4th order accuracy. Initial condition: x0=0...
  18. T

    Expansion of Taylor series for statistical functionals

    Hi By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is. My situation is that I have a functional, say \theta which...
  19. G

    Calculus II - Taylor Series Question

    Homework Statement Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) Homework Equations The Attempt at a Solution Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x) f(x) =...
  20. G

    How Accurate is the Taylor Polynomial for Approximating Sin Functions?

    New Question (Changed Old one) - Taylor Polynomial - Upper Bound for Absolute Error Homework Statement (a) Find the 3-rd degree Taylor polynomial of sin(pix) centered at x=1. (b) Use (a) to approximate sin(1.1*pi) (c) Use the remainder term to find an upper bound for the absolute error in...
  21. G

    Calculus II - Taylor Series - Error Bounds

    Homework Statement Hi, I'm really struggling with trying to come up with the error bound when doing taylor series problems Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are...
  22. V

    Examining the Taylor Series - Confused?

    hello, I'm examinating the theorem of power series, specially taylor series I know a function f(x) can be written as a series of polynomials. but using the taylor series it says that the convergence of that function is about a point a by using the Maclaurinseries a = 0 , so examinating...
  23. C

    Multi-Variable Second Order Taylor Series Expansion: Ignoring Terms

    So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...
  24. N

    Help with Taylor Series problem

    Now although this is a problem for my EE course, it is more of a calculus question so I figured I would receive the best answers by posting it in this section. I have just started on the problem but could use some input on my thoughts. So here we go (there are two parts): (problem screenshot is...
  25. morrobay

    Taylor Series for Any (x) = Function (x) for Any (x) ?

    When a Taylor Series is generated from a functions n derivatives at a single point, then is that series for any value of x equal to the original function for any value x ? For example graph the original function (x) from x= 0 to x = 10. Now plug into the Taylor Expansion for x , values...
  26. 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)...
  27. 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??
  28. 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
  29. 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 ...
  30. 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?
  31. 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...
  32. 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...
  33. I

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

    Solved Thanks guys
  34. 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...
  35. 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) =...
  36. 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...
  37. 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...
  38. 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? :$
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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...
  44. 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
  45. 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) =...
  46. 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)...
  47. 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...
  48. 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...
  49. 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?
  50. 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...
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