Taylor Definition and 878 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. 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...
  2. 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...
  3. P

    Aproximating a morse potential using a taylor polynomial

    I am not going to post my question because I want to find out how to actually use the taylor polynomial and morse potential and then apply that to my problem. Say I have to approximate the morse potential using a taylor series expanding about some value. This will then find me the force...
  4. 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...
  5. M

    Approximating accuracy of Taylor polynomials

    Homework Statement Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. e^x ≈ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} For x < 0 Homework Equations Taylor's Theorem to approximate a remainder: |R(x)| =...
  6. 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...
  7. 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...
  8. 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?
  9. H

    Find Degree 3 Taylor Polynomial Approximation of 5ln(sec(x))

    Homework Statement Find the degree 3 Taylor polynomial approximation to the function f(x)=5ln(sec(x)) about x=0. Homework Equations the taylor polynomial equation The Attempt at a Solution Here are my derivatives f(x)=5ln(secx) f'(x)=5tanx f''(x)5sec^2(x)...
  10. H

    Solve T4(x) for Taylor Polynomials of f(x)=arctan(11x)

    Homework Statement Find T4(x), the Taylor polynomial of degree 4 of the function f(x)=arctan(11x) about x=0. (You need to enter a function.) Homework Equations The taylor polynomial equation Tn(x)= f(x)+(fn(x)(x-a)^n)/n!... The Attempt at a Solution When I take every...
  11. 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...
  12. U

    Can we use Taylor expansions to find the series representation of sin(x^(3/2))?

    It is well-known that the Taylor series of \sin x about x=0 is \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} By extension, one might presume that the Taylor series of \sin x^{3/2} about x=0 is \sin x^{3/2} = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{\frac{3}{2}(2n+1)}...
  13. 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...
  14. 5

    Third degree Taylor polynomial in two variables

    Homework Statement Find the third degree Taylor polynomial about the origin of f(x,y) = \frac{\cos(x)}{1+xy} Homework Equations The Attempt at a Solution From my ventures on the Internet, this is my attempt: I see that \cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 - \cdots...
  15. 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
  16. 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}\...
  17. 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...
  18. fluidistic

    I don't understand a step in my notes, about Taylor expansion

    Homework Statement P_0 (t+dt)=P_0(t)(1-\gamma dt ) (1) Therefore P_0 (t)+\frac{dP_0 (t)}{dt} \approx P_0 (t)-\gamma P_0(t)dt. (2) Where the approximation is due to a Taylor expansion apparently. Homework Equations Taylor expansion of f around x_0 : f(x)\approx...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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)...
  24. 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...
  25. 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: $$ \displaystyle f'\left( x \right)=\sin \left( 1-\cos x \right)\sin x$$...
  26. 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...
  27. 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!}+ ...
  28. A

    Thermal: taylor series van der waals equation

    Homework Statement Show that at constant volume V and temperature T but decreasing number N=n*N_{A} of particles the Van der Waals equation of state approaches the equation of state of an ideal gas. Hint: Rearrange the equation of state into the explicit functional form P=P(v,T) and use x=1/v...
  29. F

    A question about Taylor and MacLauren series

    Hello,I have a kind of general question. So I understand that the goal of a Taylor function is to approximate a transcendental function using a polynomial function. This makes things easier to deal with sometimes. I understand that this works by chosing a polynomial function that seems to...
  30. T

    Taylor expansion of a vector field

    I was wondering if such an approximation is possible and plausible... The first term would have to look sth like this: \vec{f}(\vec{x_{0}}) + \textbf{J}_{\vec{f}}(\vec{x_{0}})\cdot(\vec{x}-\vec{x_{0}}) No clue about the second term though... We would have to calculate the Jacobian of the...
  31. R

    Mathematica Taylor Expansion in Mathematica (Multivariate)

    Hi! I'm trying to linearize a function f which is dependent of 4 variables, each one dependent of time. f[var1_,var2_,var3_,var4_]:= ... expression I use Series[f, {var1,var10 ,1},{var2,var20 ,1},...] syntax as I read on the documentation center. The problem is that the program...
  32. H

    Multivariable Calculus book with Taylor expansion for several variables

    Hi. I borrowed many multivariable Calculus book so that I can choose one for the next semester. The one I liked most is Multivariable Calculus by Ron Larson. It is full of graphics and colours, somthing that is essential to understand functions of two variables. The only thing is that it does...
  33. A

    Find the function for this Taylor series

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

    Weighted Sum in Taylor Expansion (Partial Derivatives)

    Homework Statement From step 1 to step 2, what do they mean by "Taking the weighted sum of the two squares " ? I tried and expanded everything in step 2 and it ends up as the same as step 1 (as expected), The Attempt at a Solution I tried looking up "weighted sum" and "...
  35. B

    Variational Operator/First Variation - Taylor Expansion

    Homework Statement Folks, how is the following expansion obtained for the following function ##F(x,u,u')## where x is the independent variable. The change ##\epsilon v## in ##u## where ##\epsilon## is a constant and ##v## is a function is called the variation of ##u## and denoted by...
  36. M

    What Is the Radius of Convergence for the Taylor Series of x/(1+x^2) at x=0?

    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...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. Rococo

    Trying to expand (1+x)^n using Taylor Expansion

    Homework Statement I need to use Taylor Expansion to show that: (1+x)^n = 1 + nx + n(n-1)(x^2)/2! + ...Homework Equations y(x0 + dx) = y(x0) + dx(dy/dx) + [(dx)^2/2!](d^2y/dx^2) + ... The Attempt at a Solution I've only just begun Taylor Expansion, according to my textbook I need the above...
  42. 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...
  43. T

    Where does this method of constructing the taylor expansion of arctanx fails?

    Homework Statement arctan x = ∫du/(1+u2), from 0 to x Homework Equations The Attempt at a Solution I noticed that 1/(1+u2) = 1/(1+u2)1/2 × 1/(1+u2)1/2. I decided to take the taylor series expansion of 1/(1+u2)1/2, square the result and then integrate. I got 1/(1+u2)1/2 =...
  44. 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...
  45. 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...
  46. 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...
  47. E

    Approximation Using Taylor POlynomial

    Find an approximate value of the number e-0.1 with an error less than 10-3 ı know that ex = Ʃ(from zero to ınfinity) xn / n!=1+x/1!+x 2/2!+... ı don't know how to use e-0.1 in this question.Do ı write -0.1 instead of x in ex series?
  48. 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) =...
  49. 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...
  50. 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
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