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. L

    Argumenting for this inequality involving the residual of a taylor series of ln

    Homework Statement OK I have to argument for the fact that this inequality is true, where x > 1. |R_n \ln{x}| \leq \frac{1}{n+1}(x-1)^{n+1} And I have found out that the residual is equal to this: R_n \ln{x} = \frac{1}{n!} \int^x_a{f^{n+1}(t)(x-t)^{n}dt} Homework Equations...
  2. J

    Help finding Constants for Taylor Series

    Homework Statement The Taylor expansion of ln(1+x) has terms which decay as 1/n. Show, that by choosing an appropriate constant 'c', the Taylor series of (1+cx)ln(1+x) can be made to decay as 1/n2 (assuming expansion about x=0) Homework Equations f(x)=\sum^{n=\infty}_{n=0} f(n)(0)...
  3. C

    Solving Homework Equations: Taylor Series & Beyond

    Homework Statement Homework Equations All should be there, except taylor series, which is found here: http://mathworld.wolfram.com/TaylorSeries.html The Attempt at a Solution For part a, I got: F(r)= \alpha(ke2)((-r0/r2)+(r0n/rn+1)) since force is the negative...
  4. H

    Calculate limit using taylor series

    Homework Statement Calculate: $$ \displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos \left( 1-\cos x \right)}{{{x}^{4}}}$$ Homework Equations The Attempt at a Solution Using Taylor series I have: $$ \displaystyle f'\left( x \right)=\sin \left( 1-\cos x \right)\sin x$$...
  5. H

    Taylor series of an integral function

    Homework Statement $$ \displaystyle f\left( x \right)=\int\limits_{0}^{x}{\frac{\sin t}{t}dt} $$ Calculate the Maclaurin series of third order. Homework Equations The Attempt at a Solution What I do is: $$ \displaystyle f'\left( x \right)=\frac{\sin x}{x} $$ $$ \displaystyle f''\left( x...
  6. T

    Convergence of a Taylor Series: Finding the Values of x

    Homework Statement For this problem I am to find the values of x in which the series converges. I know how to do that part of testing of convergence but constructing the summation part is what I am unsure about. I am given the follwing: 1 + 2x + \frac{3^2x^2}{2!} +\frac{4^3x^3}{3!}+ ...
  7. A

    Thermal: taylor series van der waals equation

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  8. A

    Find the function for this Taylor series

    \sum_{m=0}^\infty \frac{(m-1)^{m-1}x^{m}}{m!} Interesting result...
  9. M

    Taylor series radius converge

    Hi everybody, Firstly sorry for my bad English . I have a question related to taylor series . I did not find easy way to solve it .Derivatives are becoming more and more complex . Please help me. question : Work out the taylor series of the function x/(1+x^2) at x =0 .Find the radius of...
  10. V

    Taylor Series to Approximate Functions

    I get the many proofs behind it and all of the mechanics of how to use it. What I don't get is why it works.. What was the though process of Brook Taylor when he devised his thing? I get that each new term is literally being added to previous ones along the y-axis to approximate the y value of...
  11. D

    Reversing Taylor Series to find the original function

    Homework Statement I need to find the convergence a unknown function. Now I know the Taylor series of it which is 1/3+2/(3^2)+3/(3^3+4/(4^4+...+k/(3^k). Which mean I can just take the Riemann sum of k/(3^k) from say 0 to 50 and that would give me 3/4. However this is not enough I need...
  12. A

    MHB Finding local extrema using taylor series

    How do I find the extrema using Taylor Series?? I am so used to find extrema just by finding the first derivative (make it =0) and then finding the second derivative and then just use the formula f_xx.f_yy - f_xy and just look at the sign but this time I need to use taylor expansion. I hope you...
  13. B

    The derivative of a Taylor series?

    I took my first calculus class over the last two semesters, and my teacher and I privately worked on some harder material together. Toward the end of the school year he gave me a question that I never answered and never found an answer for. It asked me to find the derivative of a Taylor series...
  14. D

    Solving Taylor Series: (1-x^2)^(-0.5) Help

    Homework Statement Have to find the Taylor series for (1-x)^(-0.5) Then use this to find the Taylor series for (1-x^2)^(-0.5) Homework Equations The Attempt at a Solution Was able to do the expansion for the first one quite easily, but not sure how to do the second one. My initial...
  15. T

    Something I don't understand about taylor series

    Homework Statement Let's say I'm asked to find the taylor expansion for cot x, at the given point a = π/2. Homework Equations The Attempt at a Solution My first thought would be to take the mc laurin series expansion for cotx, which is: cot x = 1/x + x/3 - x3/45 ... and...
  16. S

    Taylor series expansion of a power series.

    If f(x) is a power series on S = (a-r, a+r), we should be able to expand f(x) as a taylor series about any point b within S with radius of convergence min(|b-(a-r)|, |b - (a + r)|) Does anyone have a proof of this or a link to a proof? I have seen it proved using complex analysis, but I...
  17. Sudharaka

    MHB Taylor Series Expansion Explanation

    mbeaumont99's question from Math Help Forum, Hi mbeaumont99, One thing you can do is to find the Taylor series expansion of \(f(x)=a^{x}\) and see whether it is \(\displaystyle \sum t_{n}\). The Taylor series for the function \(f \) around a neighborhood \(b\) is...
  18. J

    Taylor series of f(x)=ln(x+1) centred at 2

    Homework Statement Taylor series of f(x)=ln(x+1) centred at 2 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)4f(2) = ln(3) = 1.0986 f'(2) = 1/3 f''(2) =...
  19. P

    Complex Analysis - Radius of convergence of a Taylor series

    Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...
  20. L

    How Do You Perform the Taylor Series Expansion of e(a+x)2?

    how do you do the taylor series expansion of e(a+x)2
  21. P

    Comples analysis - Radius of convergence of a Taylor series question

    Homework Statement Find the radius of convergence of the Taylor series at z = 1 of the function: \frac{1}{e^{z}-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far. Multiply top and bottom by minus 1 to get: -1/(1-e^z) And then...
  22. B

    Approximating sin(1/10) using 3rd degree Taylor polynomial

    Homework Statement what is the 3rd degree taylor series of sin(1/10), and calculate the error of your answer. the wording of this question may be a little off, i just took a test and this was what i remembered about the question. The Attempt at a Solution i didnt think that this was...
  23. F

    MHB Exploring Taylor Series Expansions for Quadratic Equations

    Hi! I'm taking a course on Perturbation theory and as it's quite advanced the lecturer assumes everyone has a good level of maths. One of the parts is expanding roots of a quadratic equation about 0, I can understand how simple ones of the form $(1 + x)^2$ but I don't know where the answers are...
  24. 5

    Shortcut to taylor series of f, given taylor series of g

    So, I have the series of g(x) = e^{(x-1)^{2}} = 1 + (x-1)^{2} + \frac{(x-1)^{4}}{2} + \frac{(x-1)^{6}}{6} + ... + \frac{(x-1)^{2n}}{n!} and I am asked to find the series of f(x) = \frac{e^{(x-1)^{2}}-1}{(x-1)^{2}} for x \neq 1 and f(1) = 1. The Taylor series is centered about x = 1 I...
  25. K

    When does x=a in the taylor series stop being x=a?

    I'm having a hard time understanding the fundamentals of the taylor series. So I get how you continually take derivatives in order to find the coefficients but in order to do that we have to state that x=a. Well when we finally get done we have an infinite polynomial of...
  26. J

    Approximating ln(x): Taylor Series Problem Solution

    Homework Statement The first three terms of a Taylor Series centered about 1 for ln(x) is given by: \frac{x^{3}}{3} - \frac{3x^{2}}{2} + 3x - \frac{11}{6} and that \int{ln(x)dx} = xlnx - x + c Show that an approximation of ln(x) is given by: \frac{x^3}{12} - \frac{x^2}{2} +...
  27. D

    Finding a complex Taylor series

    Homework Statement Not much has gotten me in this class, and I almost want to say this has to be a typo, but I want someone else to check it out first. Homework question is that we need to show that cos(cos θ)*cosh(sin θ) = Ʃ(-1)ncos(nθ)/(2n)! for n>=0 There is a similar one involving...
  28. C

    MHB Pharaoh's Taylor series question from Yahoo Answers

    Part 1 of Pharaoh's Taylor series and modified Euler question from Yahoo Answers The Taylor series expansion about \(t=0\) is of the form: \(y(t)=y(0)+y'(0)t+\frac{y''(0)t^2}{2}+.. \)We are given \(y(0)\) and \(y'(0)\) in the initial condition, and so from the equation we have: \(y''(0) =...
  29. F

    Expanding f(x) = x/(x+1) about a=10

    Homework Statement Expand f(x) = x/(x+1) in a taylor series about a=10. Homework Equations f(x) = Ʃ (f^n(a)*(x-a)^n / n! The Attempt at a Solution I'm having a hard time arriving at the correct answer..I think I'm definitely getting lost somewhere along the way. Here's what I've...
  30. W

    Taylor Series: Can't quite work it out

    Hi Guys, Looking at some notes i have on conformal mapping and I have the following where z is complex and z* denotes its conjugate, R is a real number z* = -iR + R^2/(z-iR) and my lecturer says that using the taylor series we get, z* = -iR + iR(1+ z/iR + ...) I've been...
  31. I

    Taylor series finding sin(x^2)+cos(x) from sin(x^2) and cos(x) alone

    If I want to find the taylor series at x = 0 for sin(x^2)+cos(x)... sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! ... So why does sin(x^2) + cos(x) = 1 + x^2/2! + x^4/4! + 121x^6/6! ...? Thanks!
  32. S

    Taylor Series Remainder Theorem

    1. Prove that the MacLaurin series for cosx converges to cosx for all x. Homework Equations Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx |Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM lim(n->infinity)Rn=0 then a function is equal to its Taylor series...
  33. T

    Finding the range of validity of a taylor series

    Homework Statement I have to give the range of validity for a Taylor series built from an expression of the form: (1+(a/b)x)^c Homework Equations The Attempt at a Solution Obviously the validity does not extend to x=-(b/a) on the negative side, but should I then state that...
  34. B

    Calculating Exponent Using Taylor Series To Given Precision

    Homework Statement The course is Computational Physics, but in a sense this is a pretty straight computer science or even mathematical challenge. The first part of the assignment - the relatively easy part - was to write a Fortran program to take two variables - the number to which e...
  35. S

    What did I do wrong here? (expressing root x as taylor series about a=4)

    Homework Statement Here is the question: I don't quite know what I did wrong. My method is below. Homework Equations The Attempt at a Solution f(x)=√x f'(x)=\frac{1}{2(x)^{1/2}} f''(x)=\frac{-1}{(2)(2)(x^{3/2}} a=4 f(a)=2 f'(a)=1/4...
  36. D

    MHB Expanding Taylor Series to Get Approximate Answer

    $1+v_{t+1} = (1+v_t)\exp\left(-rv_{t-1}\right)\approx (1+v_t)(1-rv_{t-1})$ The book is linearizing the model where we generally use a Taylor Series. How was the expression expanded in the Taylor Series to get the approximate answer? Thanks.
  37. C

    Taylor series for cos[1/(1-z^2)]

    Bit stuck on this. I tried writing 1/(1-z^2) as taylor series then Cos z as taylor series, then substituting one into the other but it looked a bit dodgy. Can one simple substitute like this?
  38. C

    Taylor series of sinz-sinhz

    I have to find the first three non zero terms of this series by hand. I know the answer and it is -(z^3/3) - z^7/2520 - z^11/19958400 Which will take ages to get to by brute force. Is there a quicker way?
  39. D

    MHB Finding Taylor Series of $\dfrac{1}{z-i} \div \left(z+i\right)$

    I am trying to find the Taylor series for $$\displaystyle \dfrac{\left(\dfrac{1}{z-i}\right)}{z+i} $$ where z is a complex number.There is a reason it is set up as a fraction over the denominator so let's not move it down.
  40. R

    Alt. approach to Taylor series of derivative of arcsin(x)?

    Hi there, I was hammering out the coefficients for the Taylor Series expansion of f(x) = \frac{1}{\sqrt{1-x^2}}, which proved to be quite unsatisfying, so decide to have a look around online for alt. approaches. What I found (in addition to the method that uses the binomial theorem) was...
  41. P

    Taylor series expansion for gravitational force

    Homework Statement The magnitude of the gravitational force exerted by the Earth on an object of mass m at the Earth's surface is Fg = G*M*m/ R^2 where M and R are the mass and radius of the Earth. Let's say the object is instead a height y << R above the surface of the Earth. Using a...
  42. N

    Integration of O() terms of the Taylor series

    Hello, I have two functions say f1(β) and f2(β) as follows: f1(β)=1/(aδ^2) + 1/(bδ) + O(1) ... (1) and f2(β)= c+dδ+O(δ^2) ... (2) where δ = β-η and a,b,c,d and η are constants. Eq. (1) and (2) are the Taylor series expansions of f1(β) and f2(β) about η...
  43. A

    (Deceptively?) Simple question about Taylor series expansions

    Under what circumstances is it correct to say of the function u(x) \in L^2(-\infty,\infty) that u(x-t) = u(x) - \frac{du}{dx}t + \frac 12 \frac{d^2u}{dx^2}t^2 - \cdots = \sum_{n=0}^\infty \frac{u^{(n)}(x)}{n!}(-t)^n.
  44. S

    Does the Taylor series expansion for e^x converge quickly?

    Hello all, My question is in regards to the Taylor series expansion of f(x)=e^x=1+x+x^2/(2!)+x^3/(3!)... I calculated the value of e^(-2) using the first 4 terms, 6 terms, and then the first 8 terms. I then calculated the relative error to compare it to the true value, depcited by my...
  45. K

    What is the Taylor Series Approximation for f(x)=(x0.5-1)/0.5 and f(x)=(x-1)2?

    Homework Statement Hi! I have a couple of problems on Taylor Series Approximation. For the following equations, write out the second-order Taylor‐series approximation. Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...
  46. L

    Taylor's Theorem for Sin(a+x) and Proving Convergence | Homework Solution

    Homework Statement Taylor's theorem can be stated f(a+x)=f(a)+xf'(a)+(1/2!)(x^2)f''(a)+...+(1/n!)(x^n)Rn where Rn=fn(a+y), 0≤y≤x Use this form of Taylor's theorem to find an expansion of sin(a+x) in powers of x, and show that in this case, mod(\frac{x^n Rn}{n!})\rightarrow0 as...
  47. H

    Taylor Series of Log(z) around z=-1+i

    Homework Statement Find the taylor series of Log(z) around z=-1+i.Homework Equations The Attempt at a Solution So I have for the first few terms as \frac{1}{2}*log(2)+\frac{3\pi i}{4}+\frac{z+1-i}{-1+i}-\frac{2(z+1-i)^{2}}{(-1+i)^{2}}+\frac{3(z+1-i)^{3}}{(-1+i)^{3}}- But the correct...
  48. P

    What is the correct Taylor expansion for sin x around -pi/4 to the fourth term?

    I am asked to solve the taylor expansion of sin x around the point -pi/4 to the fourth term. I got sin(-pi/4)+cos(-pi/4)(x+pi/4)-.5sin(-pi/4)(x+pi/4)^2-1/6(cos(-pi/4)(x+pi/4)^3 but I am getting it wrong and can't see my mistake.
  49. S

    3rd order, multivariable taylor series

    Homework Statement Hello all, I have been working on a 3rd order taylor series, but the formula I have does not seem to get me the right answer. The formula I was given is for a taylor polynomial about point (a,b) is: P_3=f(a,b) +\left( f_{1}(a,b)x+f_{2}(a,b)y\right)...
  50. D

    Calculating Taylor Series for e^(x^2) around x=0

    Homework Statement Find the Taylor series of e^(x^2) about x=0 Homework Equations Taylor Series = f(a) +f'(a)(x-a) + (f''(a)(x-a)^2)/2 ... The Attempt at a Solution So, the first term is pretty obvious. It's e^0^2, which is zero. The second term is what got me...
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