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. H

    I need a method to calculate ln(x) for small x, other than Taylor series method

    I'm aiming to calculate ln(x) numerically. I'm using the following procedure for this: 1) If x is greater or equal to 1, use Newton's method. 2) If x is smaller between 0 and 1, use Taylor series expansion. Newton's method works good, but I have problems with Taylor series expansion method...
  2. jegues

    First Order Error Analysis (Taylor Series)

    Homework Statement The equation for the velocity of a falling parachutist can be computed by, v(t) = \frac{gm}{c}(1-e^{-(\frac{c}{m})t}) Use a first-order error analysis to estimate the error of v at t = 6, if g = 9.8 and m = 9.8, c = 12.5 plus or minus 1.5. Homework Equations...
  3. F

    Bounding the Error in Taylor Series Approximations for ln(1+x)

    Had a recent homework questions: Find a bound for the error |f(x)-P3(x)| in using P3(x) to approximated f(x) on the interval [-1/2,1/2] where f(x)=ln(1+x) abd P3(x) refers to the third-order Taylor polynomial. I found the Taylor series of f(x) seen below: x- x^2/2!+(2x^3)/3! I know...
  4. jegues

    Computing true percent relative error (Taylor Series)

    Homework Statement See figure attached, Homework Equations The Attempt at a Solution Isn't the Maclaurin series just simply the Taylor series around 0? (\text{i.e. } (x-c), \quad c=0, \quad x ) Also for part B, how do we go about solving for | \epsilon_{t} |? Thanks...
  5. G

    Uses of power series as opposed to taylor series

    So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...
  6. S

    Taylor Series for 1/(1+x^2) without Substitution

    Homework Statement How do we get that the Taylor Series of 1/(1+x^2) around x= 0 is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1, without using a substitution of x=-x^2 into the Taylor series for 1/(1-x)? Homework Equations The Attempt at a Solution
  7. W

    Taylor series for cartesian circle equation

    Hello. For a physics course, I need to often make use of the binomial series and it's corollary, the expansion of: \sqrt{1-x^2} This probably sounds rather stupid, but for some reason, when I do a MacClaurin expansion of this series, I cannot seem to generate the correct series, which I...
  8. K

    Taylor series of this function

    I have a homework question like this. "Find the taylor series of the function f(x) = (x2+2x+1)/(x-6)2(x+2) at x=2" I'm trying to simplify this expression so I can take the derivative. I only got this far: (x+1)(x+1)/(x-6)(x-6)(x+2) Can this be simplified more so that I can easily...
  9. E

    Finding the Taylor series of a function

    Homework Statement [sorry about the formatting, I had no idea how I would latex the sigma notation] Let f(x) = [n=1 to infinity] summation of (-1)n n2 / 3n * (x+1)n Find the Taylor series of f(x) centered at c = -1Homework Equations Taylor series defined by [n=0 to infinity] summation of...
  10. C

    Taylor Series and Maclaurin Series Help

    Homework Statement http://img704.imageshack.us/f/helpppp.png/ Homework Equations The Attempt at a Solution I know e^(x) = 1 + x + x^(2)/2! + ... But if you multiply that by (x^(4))+4x^(3)) How do you know what bn and a is?
  11. E

    What are the expected values of x for convergence of the given Taylor series?

    Hi, Im really stuck on my homework . The question is : For what values of x do you expect the following Taylor series to converge? Do not work out the series . (a) sqrtX^2-x-2 about x=1/3 b) sin(1-x^2) about x=0 for a) I've put no vlues of x would the series converge. is this correct? and...
  12. V

    Taylor Series Help: Find 1st 3 Terms at c

    Hi everybody, I hope anyone could help Homework Statement Find the first three terms of the Taylor series for f(x) at c. http://dc12.arabsh.com/i/02388/kgybq4dwkug3.png Homework Equations f(x)= f(c) + f'(c).(x-c)/1! + f"(c).(x-c)^2/2! + f'''(c).(x-c)^3/3! +...+...
  13. S

    Use Taylor series to approximate a number.

    Hello, I need help with this problem. I need to find the first three terms of the Taylor series for the function f(x)= (1 + x)^(1/3) to get an estimate for 1.06^(1/3). Hence I did: f(x)= (1 + x)^(1/3) f'(x)= (1/3)(1 + x)^(-2/3) f''(x)= (-2/9)(1 + x)^(-5/3) f(a) + f'(x)/1! * (x - a) +...
  14. jegues

    Taylor series using Geometric Series

    Let f(x) = \frac{4-4x}{4x^{2} -8x -5}; given the partial decomposition, \frac{4-4x}{4x^{2} -8x -5} = \frac{1}{5-2x} - \frac{1}{1+2x}, find the Taylor series of f(x) about 1. Express your answer in sigma notation and simplify as much as possible. Dtermine the open interval of...
  15. H

    Can a Bound be Found for the Error in Higher Order Taylor Series?

    Hello, I am trying to come up with an expression for a bound on the sum of higher order terms, above second order. Consider the following Taylor expansion of a function f(x) around a point a, f(x) = f(a) + \frac{f^{(1)}(a)}{1!}(x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2+...
  16. L

    Finding a3 in the Taylor Series for x^3ln(1+x^2)

    Homework Statement Let f(x) = x3ln(1+x2), and let the summation (from n=0 to infinity) anxn be the Taylor series for f about 0. Then what is a3? Homework Equations The Attempt at a Solution What?! I definitely don't expect the answer, but does anyone know how I could go about...
  17. P

    Taylor Series for ln(x) of Degree n at 2

    Homework Statement find Taylor polynomial for ln x of degree n, at 2 (Pn,2(x)) Homework Equations Pn,1(x)= (x-1) - (x-1)2/2 + ... + (-1)n-1(x-1)n/n The Attempt at a Solution there doesn't seem to be an obvious pattern to this. the coefficients for n=1 to n=4 are 2, -8, 24, -64...
  18. K

    Taylor Series of the inverse tangent function

    I have a shaky understanding of problems concerning Taylor Series. For example, the question below. Let f(x)=\tan^{-1}\left(\frac{1+x}{1-x}\right) where -\frac{1}{2}\leq x \leq \frac{1}{2}. Find the value of f^{2005}(0) the Taylor Series of \tan^{-1} is...
  19. J

    X Vector in 2nd Order Taylor Series Formula w/ Hessian Matrix

    The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x If you recognize this formula, do you know what the x vector is? Note: x is the x-vector, and H represents the Hessian Matrix. Thanks! The...
  20. R

    A 16-Year-Old Asks: How Can I Apply Taylor Series to Delta-F?

    Hi! I am a 16 year old trying to figure out the application of taylor series. I understand most of its uses when applied to functions like e^x, sinx, cosx, but in a mechanics book, i am required to find delta-F, a finite change in a function F. Ostensibly, this appears to be a step that needs...
  21. J

    Understanding Taylor Series Approximations

    When it says "about a point x=a", what does this mean? why not just say at x = a? Thanks
  22. C

    Finding the Taylor Series of f(x) = x/(2+x)

    Homework Statement Obtain the Taylor series in powers of x + 1 for f(x) = x/(2 + x), giving the general term. Homework Equations The Attempt at a Solution Wrote it out as x*(1/1-(-(x+1)).
  23. J

    Taylor Series Expansion - Don't understand how to use

    Homework Statement This is actually not a problem, it's something in my notes. The function I am supposed to be approximating is V(x) = V0(1 - ex/a)2 - V0 V0 and a are constants. Homework Equations The Attempt at a Solution It says that the function given is not a parabola. But it can be...
  24. D

    F(x) of a taylor series that looks a lot like an exponential

    Hello, I am trying to evaluate the series \sum{\frac{x^n}{n!}e^{cn^2}} where c is a constant. I think this problem is equivalent to find f(x) such that \frac{d^{n}f(0)}{dx^{n}} = \frac{e^{cn^{2}}}{n!} I believe this must be a modified exponential since for c=0, it reduces to...
  25. jegues

    Taylor Series using Geometric Series and Power Series

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...
  26. D

    How Does the Taylor Polynomial Change When Centered at a Non-Zero Point?

    Hello, if I understand correctly the Taylor approximation for a=0 gives me the possibility to approximate a function, say sin(x), at any x. But, what gives me Taylor polynomial at some point http://latex.codecogs.com/gif.latex?a\neq0 ,[/URL] what's the difference? what does it mean centred...
  27. A

    Taylor Series Linearization of f(x) Around x0

    I am trying to linearize a function, f(x), where x is a normally distributed N(0,1) random variable. How can I perform a taylor series expansion around a deterministic value x0? Thanks.
  28. W

    Deriving taylor series for v/c and gamma

    Homework Statement The velocity of a proton relative to our galaxy is vp/c = 1-(0.5*10^20), i.e. almost one. Such protons are actually observed. When velocity it very nearly one \gamma is very large. 1/\gamma is very small. Use Taylor series to show that for v almost one we have...
  29. G

    Taylor Series in Multiple Variables

    Can anyone help me for the leading order terms in the taylor series for the function f(x,y) = Sqrt(a*x^8+b*x^4*y^4+y^8), centered at x=0,y = 0 and a,b,c constants?
  30. Z

    Convergence of Taylor series in several variables

    where do a multiple Taylor series converge ?? i mean if given a function f(x,y) can i expand this f into a double Taylor series that will converge on a rectangle ? for example , if one can ensure that it converges for |x| <1 and |y| <1
  31. A

    Derive Multivariable Taylor Series

    Hello all, I am currently studying multivariable calculus, and I am interested in the Taylor series for two variable function. I am not sure where to begin; I cannot understand any of the proofs (which are apparently sparse) on the internet; they all just state it using a sigma sum; not...
  32. W

    Proving Cauchy's Inequality for Analytic Functions with Distance Constraints"

    Homework Statement Let f(z)=\sum_{n=0}^{\infty} a_n z^n be analytic at {z: |z|<R} and satisfies: |f(z)| \leq M for every |z|<R. Let's define: d=the distance between the origin and the closest zero of f(z). Prove: d \geq \frac{R|a_0|}{M+|a_0|} . Hope you'll be able to help me...
  33. S

    Taylor series radius of convergence

    Hi, We need a generic expression of a taylor series nth term to find out the radius of convergence of the series. However, there are series where I don't think it is even possible to find a generic term. How do we find the radius of convergence in such cases? e.g. sqrt (1 - x^2) There...
  34. Saladsamurai

    Partial Taylor Series Expansion

    "Partial" Taylor Series Expansion It has been awhile since I have had to use a Taylor series expansion (from scratch). I looked it up on wiki and the rules are easy enough, I am just a little confused as to how I apply it to a multivariable function, but only expand it about one variable...
  35. Z

    Understanding Taylor Series and Error Bounds in Calculus

    I'm doing some review over summer before starting college, and one of the practice exams has a question pertaining to the remainder of a taylor series Homework Statement Show that \left|\cos{(1+x)}-\{\cos{(1)}(1-\frac{x^2}{2})-\sin{(1)}(x-\frac{x^3}{3!})\}\right|<\frac{1}{15000} for |x|<0.2...
  36. P

    Taylor Series Expansion About the Point i

    Taylor Series Expansion About the Point "i" Homework Statement Calculate the radius of convergence of the Taylor series for \frac{1}{z^2-2z+2} about the point i. The Attempt at a Solution I can find the radius of convergence if I can determine the expansion but I can't seem to...
  37. N

    Taylor series with plus inside

    i can't understand how the got this variation of taylor series formula f(x+h)=\sum_{k=0}^{\infty}\frac{f^{(k)}(x)}{k!}(h)^k http://mathworld.wolfram.com/TaylorSeries.html when around some point there is no x-x_0
  38. T

    Calculating errors in Functions of two variables Taylor Series

    Homework Statement From the taylor series we can replace x =x_{0} + h but how does \delta f = f(x_{0} + h, y_{0} + k) - f(x_{0},y_{0}) become \delta f = hf(x_{0}, y_{0}) + kf(x_{0}, y_{0}) I can see the first step, but how do you get it to the second?Homework Equations The Attempt at a Solution
  39. N

    Taylor Series for sin(x) Centered at π/2 with Infinite Radius of Convergence

    f(x)=sinx taylor series centered at pi/2 sum((-1)^n (x-pi/2)^(2n)/(2n)! , n=0,infty ) with radius of convergence infty
  40. N

    Is this a correct taylor series representation centered at 1

    f(x)=1/(1-x^2)^(1/2) 1/x^(1/2)=1+ sum(( (-1)^n 1*3*5*7...(2n-1)(x-1)^n )/(2^n n! ) , n=1, infty ) thus 1/(1-x^2)^(1/2) = 1+ sum(( 1*3*5*7...(2n-1)(x^2)^n )/(2^n n! ) , n=1, infty ) is this a correct taylor series representation centered at 1
  41. J

    Kinematic equation looks like a Taylor series

    I was just pondering today how the kinematic equation for position looks like a taylor expansion. x = x0 + dx/dt *t + (1/2)*d2x/dt2*t2 I believe it continues like that, exactly like a taylor expansion does, so the next term would be (1/6)*d3x/dt3*t3 If it is indeed a taylor expansion, what...
  42. S

    Finding the Maximum Remainder in a Taylor Series: Explained

    Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series? I read the wiki article and didn't help me at all, http://en.wikipedia.org/wiki/Taylor's_theorem My book talks about something like this(image is...
  43. R

    Taylor series expansion for xln(x) with x = 1

    Homework Statement For f(x) = xln(x), find the taylor series expansion of f(x) about x = 1, and write the infinite series in compact form. 2. The attempt at a solution I can find the expansion itself fine, these are the first few terms: 0 + (x-1) + \frac{(x-1)^{2}}{2!} -...
  44. A

    Using Taylor Series to Approximate Force in Gravitational Fields

    so F = mgR2/(R+h)2 where R is the radius of the earth. consider the situation where h is much smaller than R. a) show that F is approximately equal to mg b)express F as mg multiplied by a series in h/R so i need help on getting started. would showing that F is approximately equal...
  45. A

    Finding Taylor Series for f(x) = $\frac{x^2+1}{4x+5}$

    Homework Statement find the taylor series for the function f(x) = \frac{x^2+1}{4x+5} Homework Equations N/A The Attempt at a Solution how to do this? 1st attempt. i did turn it this term \frac{x}{4} + \frac{-5x+4}{16x+20} can i turn this to taylor series? maybe i know how to make...
  46. S

    Revelation about Taylor series and linear/quadratic approximations

    I don't have anyone else to ask. So I have to ask you guys. I learned about Taylor series, and then I went back and looked at linear and quadratic approximations, and they are Taylor series except only taken so far. I'm pretty much just looking for confirmation on my idea, it seems perfect.
  47. A

    Finding Taylor Series for (x-1)/(1+x) at x=1

    Homework Statement find taylor series for \frac{x-1}{1+x} at x=1 Homework Equations The Attempt at a Solution how to change this form \frac{x-1}{1+x} to something like this \frac{1}{1+a} or \frac{1}{1-a} help me please T_T or should i do like this \sum\frac{f^n(1)(x-1)^n}{n!} and find...
  48. J

    Taylor Series of 1/w: Proving Convergence

    Homework Statement Find the Taylor Series for f(w) = 1/w centered at w0 = 1 using 1/w = (1/1 + (w-1)). Show that the series converges when |w-1| < 1 Homework Equations use 1/w = (1/1 + (w-1)) The Attempt at a Solution
  49. H

    Taylor Series question about error:

    Homework Statement This is a three part question: It is based off the first two sections. I'm pretty sure the first two answers are correct, but I have no idea how to do the third question. Write the First three nonzero terms and the general term of the Taylor series expansion about x=0...
  50. B

    Derivative of a Taylor Series f(x) is unknown

    Homework Statement If \sum_{n=0}^{\infty} a_{n}x^n is a Taylor series that converges to f(x) for all real x, then f'(1) = ? Homework Equations A Taylor series: \sum_{n=0}^{\infty} \frac {f^{(n)}(c)}{n!}(x-c)^n and the dirv of a Taylor series: f'(x)=\sum_{n=0}^{\infty}...
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