Taylor series Definition and 492 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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. T

    Taylor series Mostly conceptual

    I was just curious why when doing a taylor series like xe^(-x^3) we must first find the series of e^x then basically work it from there, why can't we instead do it directly by taking the derivatives of xe^(-x^3). But doing it that way doesn't give a working taylor series why is this so?
  7. K

    What is the difference between power series and Taylor series?

    My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!:smile: Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1...
  8. F

    Very Easy Taylor Series Approximation Help

    Homework Statement Approximate f by a Taylor polynomial with degree n at the number a. f(x) = x^(1/2) a=4 n=2 4<x<4.2 (This information may not be needed for this, there are two parts but I only need help on the first) Homework Equations Summation f^(i) (a) * (x-a)^i / i! The Attempt at...
  9. I

    Taylor Series: Equivalence of Two Forms Explained

    I don't get how these two forms of the taylor series are equivalent: f(x+h)= \sum_{k=0}^{\infty} \frac{f^k(x)}{k!} h^k f(x) = \sum_{k=0}^{\infty} \frac{f^k(0)}{k!}x^k The second one makes sense but I just can't derive the first form using the second. I know its something very simple...
  10. T

    Taylor Series for exp(x^3) around x = 2

    Homework Statement Give the Taylor Series for exp(x^3) around x = 2. Homework Equations f(x) = Sum[f(nth derivative)(x-2)^n]/n! The Attempt at a Solution I know the solution for e^x but can't seem to find a formula for the nth derivative of exp(x^3) around x = 2. Thanks for...
  11. T

    Taylor Series Help: ƒ(x) = e^(x/2), g(x) = ((e^(x/2)) - 1)/x

    Let ƒ be the function given by f (x) = e ^ (x / 2) (a) Write the first four nonzero terms and the general term for the Taylor series expansion of ƒ(x) about x = 0. (b) Use the result from part (a) to write the first three nonzero terms and the general term of the series expansion about x = 0...
  12. D

    How Does Decomposing a Complex Taylor Series Help Analyze Its Behavior?

    TASK: Assuming a complex function f(z) can be expanded as a Taylor series around z=0, i.e.: f(z)=\sum_{n=0}^{\infty}a_{n}z^n Setting z=r*exp(i*theta), assuming a_n is real, find real part u(r, theta), imaginary part v(r,theta). Comment the result, especially for r=1. MY SOLUTION...
  13. T

    Can the Taylor Series of Analytic Functions be Proven?

    i'm having a hard time understanding taylor series and why it works and how it works. if someone could please explain it to me that would be great. My teacher explained it in class but he goes so fast that i have no idea what he's saying. he did give us some practice problems but if i have no...
  14. M

    Understanding the Taylor Series Concept: Exploring its Uses and Applications

    hi everyone, I am just learning the taylor series at school. I am slightly confused. in my textbook, one of hte exercises is to find hte nth degree taylor polynomial of x^4 about a=-1. n is 4 in this case so this gives me a long polynomial. i understand that inputting any x value into this...
  15. V

    How Does Taylor Series Approximation Determine Electric Potential Over Distance?

    The electric potential V at a distance R along the axis perpendicular to the center of a charged disc with radius a and constant charge density d is give by V = 2pi*d*(SQRT(R^2 +a^2) - R) Show that for large R V = pi*a^2*d / R This is what I have done so far... V = 2pi*d *...
  16. T

    How Can Taylor Series Approximate Second Derivatives?

    I am supposed to prove using taylor series the following: \frac{d^2\Psi}{dx^2} \approx \frac{1}{h^2}[\Psi (x+h) - 2\Psi(x) + \Psi (x-h)] where x is the point where the derivative is evaluated and h is a small quantity. what i have done is used: f(x+h)= f(x) + f'(x) h +...
  17. K

    Expanding f(x,y) with Double Taylor Series

    Let be an analytic function f(x,y) so we want to take its Taylor series, my question is if we can do this: -First we expand f(x,y) on powers of y considering x a constant so: f(x,y)= \sum_{n=0}^{\infty}a_{n} (x)y^{n} and then we expand a(n,x) for every n into powers of x so we have...
  18. C

    Finding the 2005th Derivative with Taylor Series for Inverse Tan Function

    I have got a question here that puzzles me. How do I use TAYLOR SERIES to find the 2005th derivative for the function when x=0 for the following function: f(x) = inverse tan [(1+x)/(1-x)] Part (1) I was hinted that differentiating inverse tan x is = 1/(1+x^2). Part (2) After which, I need to...
  19. C

    Taylor Series Approximation for Solving Initial Value Problems

    With a simple ODE like \frac{ds}{dt} = 10 - 9.8t and you're given an initial condition of s(0) = 1, when doing the approximation would s'(0) = 10 - 9.8(0), s'' = ... etc?
  20. A

    Starting a Taylor series problem, .

    Compute the Taylor series for f(x)= sq root (x) about x=1. Determine where the series sconverges absolutely, converges conditionally, and diverges. Hint: 2(k!)=2*4*6...(2k-2)*2k. Also 1<2, 3<4, 5<6,..., 2k-1<2k should help you out with a comparision.
  21. E

    Complex analysis taylor series Q

    hi, I'm wondering if someone can help me out with this question: "What are the first two non-zero terms of the Taylor series f(z) = \frac {sin(z)} {1 - z^4} expanded about z = 0. (Don't use any differentiation. Just cross multiply and do mental arithmetic)" I know the formula for...
  22. T

    Taylor Series for ln(1-x): Get Help Now

    may i know from ln(1-x) how to become - [infinity (sum) k=1] x^k / k ? pls help
  23. P

    Taylor series homework problem

    Dear friends, I have a question on a taylor series, that is this one: A·e^(i (x)) That is: cos (x)+ i sin (x) becouse of the taylor's. But, is this wrong? A·e^(v (x)) = cos (x)+ v sin (x) (v is a vector). Tks.
  24. A

    Taylor Series to x places of decimals

    Hi all, here's the problem: given: tan^(-1)= x - x^3/3 + x^5/5 using the result tan^(-1) (1)= pi/4 how many terms of the series are needed to calculate pi to ten places of decimals? note: this is supposed to say tan^(-1) and tan^(-1)[1] respectively Does anyone know whether...
  25. N

    What did I do wrong in my Taylor Series Expansion for y=kcosh(x/k)?

    y=kcosh(x/k) = k(e^(x/w) + e^(-x/k)) y ~ k[1 + x/k + (1/2!)(x^2/k^2) + (1/3!)(x^3/k^3)] +...+ k[1 - x/k +(1/2!)(x^2/k^2) - (1/3!)(x^3/k^3) +...] All odd terms except 1 cancel out. So we are left with y = k [2 + (2/2!)(x^2/k^2) + (2/4!)(x^4/k^4) + (2/6!)(x^6/k^6) +...] I've been...
  26. A

    How Can You Derive the Taylor Series of 1/sqrt(cosx) Using the Series for cosx?

    Is there a way to get the Taylor series of 1/sqrt(cosx), without using the direct f(x)=f(0)+xf'(0)+(x^2/2!)f''(0)+(x^3/3!)f'''(0)... form, just by manipulating it if you already know the series for cosx?
  27. P

    Solving Differential Equations with Simple Taylor Series Method

    Hi, I was reading this math book once... and it had a method for solving differential equations of 1st (And maybe 2nd? I don't remember) order by using simple Taylor series... I didn't even have to understand much of what was going on, except that I followed some simple rule and I ended up...
  28. B

    Understanding Taylor Series Error & Degrees of Variables

    Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...
  29. B

    Exploring the Relationship between Taylor Series Degree & Error

    Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...
  30. B

    Solving Taylor Series Problem with m-th Derivative Bound

    Hi , I have some difficulties to solve this problem. It is from my numerical methods class but the problem is about taylor series: It is known that for 4 < x < 6, the absolute value of the m-th derivative of a certain function f(x) is bounded by m times the absolute value of the quadratic...
  31. M

    Taylor series expansion question

    Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...
  32. X

    Taylor Series Expansion for z^i at z=1+i: First Three Terms

    I need to find the first three terms of this series. Am I correct in saying z^i = exp(i*Log(z)), then using the taylor series for e^z, giving me: (i*Log(z)) - 1/2*(Log(z))^2 - i/6*(Log(z))^3 + 1/24*(Log(z))^4 + ... I haven't worked it out, but this seems to mean that the coefficients...
  33. R

    Taylor Series for Showing B_{x}(x+dx)-B_{x}(x) Approximation

    How do i show that B_{x}(x+dx,y,z)-B_{x}(x,y,z)\approx \frac{\partial B_{x}(x,y,z)}{\partial x} dx using a Taylor series to the first term. Using a Taylor series does B(x) = B(a) + B'(a)(x-a)? In that case what would B(x+dx) be and how can i obtain the desired result from this? Thanks in...
  34. P

    Finding Taylor Series of Functions - Tips to Make it Easier

    I was wondering if someone can give me some tips for finding the taylor series of functions. For example this was a test question we had: Find the taylor series of f(x)=ln(x) about x=e I know how to start it off but I get confused halfway through and can't seem to figure out what to do...
  35. A

    How Is the Full Taylor Series Derived Beyond Its Linear Approximation?

    I understand what a linear approximation, and how it is derived using the point-slope formula: f(x)\approx f(a)+f'(a)(x-a) These are the first three terms of a Taylor series, so I was wondering how the rest was derived? Thanks for your help.
  36. D

    Taylor series exp. & a couple of other questions

    Problem Find the sum of the series s(x) = \sum _{n=1} ^{\infty} \frac{1}{2^{n}} \tan \frac{x}{2^n} Solution If s(x) = \sum _{n=1} ^{\infty} \frac{1}{2^{n}} \tan \frac{x}{2^n} = \frac{x}{3} + \frac{x^3}{45} + \frac{2x^5}{945} + \dotsb \cot x = \frac{1}{x} - \frac{x}{3} -...
  37. D

    Taylor Series Problem Solved: Coefficient of x^7

    Help me out with this Taylor series problem: The Taylor series for sin x about x = 0 is x-x^3/3!+x^5/5!-... If f is a function such that f '(x)=sin(x^2), then the coefficient of x^7 in the Taylor series for f(x) about x=0 is? thanks
  38. A

    Dummy Variables in Taylor Series

    Taylor Series in x-a Hi, I've got a question about the use of dummy variables in Taylor Series. We are asked to expand: g(x) = xlnx In terms of (x-2). So originally, I used a dummy variable approach to try and find an answer. Let t = x-2, so x = t+2. g(x) = (t+2)ln(2+t)...
  39. dduardo

    Taylor Series Solution for y'' = e^y at x=0, y(0)=0, y'(0)=-1

    This problem has been bugging me and I can't seem to figure it out: y'' = e^y where y(0)= 0 and y'(0)= -1 I'm supposed to get the first 6 nonzero terms I know the form is: y(x) = y(0) + y'(0)x/1! + y''(0)x^2/2! + y'''(0)x^3/3! +... and I know the first two terms are y(x) = 0 -...
  40. T

    Taylor Series for ln(x+1) at x=1

    Find the 4th term of the Taylor series centerd at x=1 for f(x)=ln(x+1) f(x)=ln(1=x) f'(x)=(1+x)^-1 f"(x)=(-1)[(1+x)^-2] f"'(x)=(2)[(1+x)^-3] f""(x)=-6[(1+x)^-4)] Plug in 1: .6931 .5 -.25 .25 -.375 What do I do next? (Also, is the 4th term the 4th term starting with f(x)? or the...
  41. C

    Taylor series with partial derivatives

    We were gievn a question in tutorial last week asking us to calculate the Taylor series of the function f(x,y) = e^(x^(2) + y^(2)) to second order in h and k about the point x=0, y=0 I've got the forumla here with all the h's and k's in it and have it written down, but it's actually how to...
  42. K

    What is the Taylor series for i^i?

    I'm having some problems expanding i^i, could anyone help? I know it becomes a real number somehow, and I'm familiar with the e^(i * pi) expansion, but is the i^i done in the same way?
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