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. C

    Taylor Series: Simple Homework Statement, Find 1st & 2nd Terms

    Homework Statement "Determine the first two non-vanishing terms in the Taylor series of \frac{1-\cos(x)}{x^2} about x = 0 using the definition of the Taylor series (i.e. compute the derivatives of the function)." So I know how compute the Taylor series about x=0; it involves finding f(0)...
  2. F

    Indeterminate forms of Taylor series

    Can someone please explain how the taylor series would work if x, the given value from the function, is equal to a, the value at which you expand the function? For example, let's take 1/(1-x) as an example. The taylor series for this with a=0 is Ʃ(n from 0 to infinity) x^n. But if we let...
  3. O

    What should i do after using Taylor series?

    Homework Statement The first equation on the uploaded paper converts to the last equation.Homework Equations when i substitute ln (1-u)=-u-(1/2)(u^2) into the first equation, i can get the first term in (3rd equation). but the second term of the 3rd equation ?The Attempt at a Solution I tried...
  4. C

    Functional or regular (partial) taylor series in Field theory

    When expanding a function (for example the determinant of the space-time metric g) as a functional of a perturbation from the flat metric ##h_{\mu \nu}##, i.e. ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ## i would think that the thing to do is to recognize that ##g_{\mu \nu}## and thus also...
  5. S

    MHB Taylor series: Changing point of differentiation

    Continuing from http://www.mathhelpboards.com/f10/taylor-series-x-%3D-1-arctan-x-5056/: The discussion in that thread gave rise to a general question to me: Does not the point of differentiation change when one makes the substitution h = x -a? I like Serena affirmed this "conjecture but...
  6. S

    MHB Taylor series at x = 1: arctan(x)

    Hey forum. Is there any way one can take advantage of the Maclaurin series of \arctan (x) to obtain the Taylor series of \arctan (x) at x = 1? I attempted to obtain the series in the suggested manner but to no avail. We have \arctan (x) = \sum_{n=0}^\infty \frac { f^{(n)}(1) }{n!} (x - 1)^n...
  7. DeusAbscondus

    MHB How to use Taylor series to represent any well-behaved f(x)

    Does one assess $x$ at $x=0$ for the entire series? (If so, wouldn't that have the effect of "zeroing" all the co-efficients when one computes?) only raising the value of $k$ by $1$ at each iteration? and thereby raising the order of derivative at each...
  8. Petrus

    MHB Calculate Taylor Series of f^{(18)}(0)

    Calculate f^{(18)}(0) if f(x)=x^2 \ln(1+9x) if we start with ln(1+9x) and ignore x^2 we can calculate that f'(x)= \frac{9}{1+9x} <=> f'(0)=9 f''(x)= \frac{9^2}{(1+9x)^2} <=> f'(0)=9^2 . . . f^{n}(x)= \frac{9^n}{(1+9x)^n} <=> f'(0)=9^n how does it work after? Don't we have to use product rule...
  9. Petrus

    MHB Tackling Taylor Series with f(x)=sin(x^3)

    Hello MHB, I am working with Taylor series pretty new for me, I am working with a problem from my book f(x)=\sin(x^3), find f^{(15)}(0). I know that \sin(x) = 1-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}...Rest How does this work now =S? Regards, |\pi\rangle
  10. B

    Evaluate the antiderivative as a Taylor Series

    Homework Statement Evaluate the anti derivative ∫e^x^2 dx as a Taylor Series Homework Equations \frac{f^(n)(a)}{n!}(x-a)^n The Attempt at a Solution Where do I start, I am not sure I understand the question
  11. A

    Taylor series, an intuitive view

    Homework Statement I read that the taylor series was a way to approximate the a function f(x) graphically, by addition and subtraction. So say I have \frac{1}{1-x}=1+x+x^{2}+x^{3}+...+x^{n}... suppose x=3, then the left and right side of the equation can't possibly equal the same thing...
  12. Fernando Revilla

    MHB Don's question at Yahoo Answers (Taylor series)

    Here is the question: Here is a link to the question: How to Find the Taylor series for the function f(x)=(1)/(x) centered at a=-3? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  13. L

    Find Taylor series generated by e^x centered at 0.

    1. a. Find Taylor series generated by ex2 centered at 0. b. Express ∫ex2dx as a Taylor series. 2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series: ex: 1 + x + x2/2! + x3/3! + ... ex2: 1 + x2 + x4/2! + x6/3! + ... For part b...
  14. S

    Why is the correction important?

    Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...
  15. S

    Taylor Series of f(x) = 1/(1-6x) at c=6

    Homework Statement Find the Taylor Series for f(x) = 1/(1-6) centered at c=6 Homework Equations ∞ Ʃ Fn(a)(x-a)/n! n=0 The Attempt at a Solution I believe that the nth derivative of 1/(1-6x) is (-6)n-1n!/(1-6x)n+1 So i figured that the taylor series at c=6 would be...
  16. X

    Understanding Taylor Series for Solving Complex Equations

    Homework Statement f(x)=\frac{4x}{(4+x^{2})^{2}}Homework Equations \frac{1}{1-x} = \sum x^{n} The Attempt at a Solution How am I supposed to use that equation to solve the main problem. I have the solution but I don't understand how to do any of it. My professor is horrible, been on...
  17. S

    Understanding the Taylor Series of e^x/(x-1)

    Homework Statement Let g(x) = \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} \frac{B_n}{n!} x^n be the taylor series for g about 0. Show B_0 = 1 and \sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 .Homework Equations The Attempt at a Solution g(x) = \sum_{n=0}^{\infty} \frac{g^{(n)}(0)}{n!} x^n , but...
  18. STEMucator

    Integrating Taylor Series for Sine Functions

    Homework Statement A problem from advanced calculus by Taylor : http://gyazo.com/5d52ea79420c8998a668fab0010857cf Homework Equations ##sin(x) = \sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{(2n+1)!}## ##sin(3x) = \sum_{n=0}^{∞} (-1)^n \frac{3^{2n+1}x^{2n+1}}{(2n+1)!}## The Attempt at a Solution...
  19. STEMucator

    Calculating Taylor Series Expansion for ##f(x)##

    Homework Statement Calculate the Taylor series expansion about x=0 as far as the term in ##x^2## for the function : ##f(x) = \frac{x-sinx}{e^{-x} - 1 + ln(x+1)}## when ##x≠0## ##f(x) = 1## when ##x=0## Homework Equations Some common Taylor expansions. The Attempt at a Solution...
  20. T

    Is there any benefit to using Taylor series centered at nonzero value

    over a Maclaurin series? Also, how do I calculate e^0 using Maclaurin series? I'm getting 0^0.
  21. S

    Why Taylor Series works so well for some functions and not for others

    About a week ago, I learned about linear approximation from a great youtube video, it was by Adrian Banner and the series of his lectures I think were from his book Calculus LifeSaver. I truly thought it was so beautiful and powerful a concept. Shortly I also got to know the Taylor Series and...
  22. E

    Taylor series for getting different formulas

    I am trying to establish why, I'm assuming one uses taylor series, \frac{\partial u}{\partial t}(t+k/2, x)= (u(t+k,x)-u(t,x))/k + O(k^2) I have tried every possible combination of adding/subtracting taylor series, but either I can not get it exactly or my O(k^2) term doesn't work out (it's...
  23. I

    MHB Proving $f(x_k+εp)<f(x_k)$ with Taylor Series

    Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, but what's then? (Taylor series: $f(x_0+p)=f(x_0)+p^T▽f(x_0)+(1/2)p^T▽^2f(x_0)p+...$ => here...
  24. F

    Taylor series expansion of Dirac delta

    I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...
  25. I

    Finding a Taylor Series from a function and approximation of sums

    Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...
  26. J

    Taylor Series Problem - Question and my attempt so far

    Question: http://i.imgur.com/GsjeL.png Here is my attempt so far: http://i.imgur.com/AyOCm.png Note: I've used m where the question has used j. My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to...
  27. C

    Taylor series question

    The Taylor Series of f(x) = exp(-x^2) at x = 0 is 1-x^2... Why is this? The formula for Taylor Series is f(x) = f(0) + (x/1!)(f'(0)) + (x^2/2!)(f''(0)) + ... and f'(x) = -2x(exp(-x^2)) therefore f'(0) = 0? Can someone please explain why it is 1-x^2?
  28. C

    Understanding and Solving the Taylor Series for a Specific Point

    What does it mean to calculate the Taylor series ABOUT a particular point? I understand the formula for the Taylor series but how do you solve it about a particular point for a function? It's the about the particular point that confuses me. Could someone please explain this and provide...
  29. M

    Why the Taylor Series has a Factorial Factor

    Why in Taylor series we have some factoriel ##!## factor. f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+... Why we have that ##\frac{1}{n!}## factor?
  30. T

    On Taylor Series Expansion and Complex Integrals

    I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...
  31. F

    Compute Tricky Limit Using Taylor Series and De L'hopital's Theorem

    Homework Statement compute the following limit: ## \displaystyle{\lim_{x\to +\infty} x \left((1+\frac{1}{x})^{x} - e \right)} ## The Attempt at a Solution i wanted to use the taylor expansion, but didn't know what ##x_0## would be correct, as the x goes to ## \infty##. also, i tried to...
  32. P

    How to find the cosine of i using Taylor series?

    Is there a way to find the cosine of i, the imaginary unit, by computing the following infinite sum? cos(i)=\sum_{n=0}^\infty \frac{(-1)^ni^{2n}}{(2n)!} Since the value of ##i^{2n}## alternates between -1 and 1 for every ##n\in\mathbb{N}##, it can be rewritten as ##(-1)^n##...
  33. H

    How to write taylor series in sigma notation

    Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...
  34. P

    Modifying taylor series of e^x

    I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course: \sum_{n=0}^\infty \frac{x^n}{n!} would change the equation. How would changing this to: \sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)} change the equation? Would it still be...
  35. R

    Polylogarithm and taylor series

    let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series: Lin(x)= Ʃ xk/kn 1.calculate the radius of convergence [b]3. when i attempted this part, i couldn't use theratio or root test, so by comparison i got R=∞...
  36. O

    MHB Taylor Series: Exploring Properties & Applications

    Hello Everyone! Suppose $f(x)$ can be written as $f(x)=P_n(x)+R_n(x)$ where the first term on the RHS is the Taylor polynomial and the second term is the remainder. If the sum $\sum _{n=0} ^{\infty} = c_n x^n$ converges for $|x|<R$, does this mean I can freely write $f(x)=\sum _{n=0} ^{\infty}...
  37. E

    Approximation sin(x) taylor Series and Accuracy

    Homework Statement One uses the approximation sin(x) = x to calculate the oscillation period of a simple gravity pendulum. Which is the maximal angle of deflection (in degree) such that this approximation is accurate to a) 10%, b) 1%, c) 0.1%. You can estimate the accuracy by using the next...
  38. P

    MHB Finding $a_n$s for $f(z)$ in a Taylor Series

    Consider the function $$f(z)=e^{\frac{1}{1-z}}$$ It has an essential singularity at $z_0=1$ and hence it can be expanded in a Laurent series at $z_0$. But I'm interested in Taylor expansion. The function is analytic in the unit open disc at the origin, so I'm looking for $a_n$ where...
  39. A

    A question about Taylor Series

    Find the Taylor series for cosx and indicate why it converges to cosx for all x in R. The Taylor series for cosx can be found by differentiating sum_{k=0}^{\infty} \frac{(-1)^k (x^{2k+1})}{(2k+1)!} on both sides... But I'm not sure what the question means by "why it converges to cosx for...
  40. Y

    Taylor Series about exp(-1/x^2)

    Homework Statement Homework Equations We just learned basic Taylor Series expansion about C, f(x) = f(C) + f'(C)(x - C) + [f''(C)(x - C)^2]/2 + ...The Attempt at a Solution Well the previous few questions involved finding the limit of the function and the derivative of the function as X...
  41. W

    Taylor series error term - graphical representation

    Hello all, Recently I've found something very interesting concerning Taylor series. It's a graphical representation of a second order error bound of the series. Here is the link: http://www.karlscalculus.org/l8_4-1.html My question is: is it possible to represent higher order error bounds...
  42. P

    Find the Taylor Series for 28^(3/5) Up to First Order – Tips & Suggestions

    Hello, What polynomial should I use for finding Taylor series for 28^(3/5) up to the first order? I mean, aside x^3 around 2. Any suggestions?
  43. T

    Taylor Series, working backwards

    Homework Statement Okay, first there is an explanation of the Taylor Series equation. This I don't have a problem with. Then, we have this: Consider the power series 2 - (2/3)x + (2/9)x^2 - (2/27)x^3. What rational function does this power series represent? Homework Equations / The Attempt...
  44. Square1

    Taylor Series Interval of COnvergence and Differention + Integration of it

    OK... "A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence" When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact? Does this come into play later in a...
  45. B

    Would this require a Taylor Series Proof

    Show that abs[ sin (x) - 6x/(6+x^2) ] <= x^5/24, for all x in [0,2] I tried to use the sine function taylor expansion but I get stuck
  46. M

    Two Variable 2nd Order Taylor Series Approximation

    Homework Statement Derive the Derive the two variable second order Taylor series approximation, below, to f(x,y) = x^3 + y^3 – 7xy centred at (a,b) = (6,‐4) f(x,y) ≈ Q(x,y) = f(a,b) + \frac{∂f}{∂x}| (x-a) + \frac{∂f}{∂x}|(y-b) + \frac{1}{2!}[\frac{∂^2f}{∂x^2}| (x-a)^2 + 2\frac{∂^2f}{∂x∂y}\...
  47. X

    How do I sum up a Taylor series with unusual coefficients?

    I need to calculate \sum_{n=0}^{∞}x^{(2^n)} for 0≤x<1. It doesn't resemble any basic taylor series, so I have no idea how to sum it up. Any hint, or the resulting formula? This series comes from a physical problem, so I suppose (if I didn't make a mistake) that the series is sumable, and...
  48. 1

    Questions regarding Taylor series

    We just had a lecture on power series today (Taylor and McLaurin's) and I had a couple of questions: What does it mean for an expansion to be "around the origin"? I thought that the expansion provided an approximation to the original function at all points for which the function was defined...
  49. C

    Taylor series expansion for gravitational potential energy. GMm/r=mgh near the earth

    Using taylor series expansion to prove gravitational potential energy equation, GMm/r=mgh at distances close to the earth. R= radius of the Earth h= height above surface of the Earth m= mass of object M= Mass of the earth U = - GmM/(R + h) = - GmM/R(1+ h/R) = - (GmM/R)(1+ h/R)^-1 do a...
  50. S

    Taylor Series and Maclaurin Series Doubt

    Homework Statement If I take a function f(x) and its taylor series, then will the infinite series give me the value of the function at any x value or will it only give proper values for x≈a? For example, If I take a maclaurin series for a function will it give me proper values for all x...
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