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 to estimate sums

    [b]1. Use Taylor's expansion about zero to find approximations as follows. You need not compute explicitly the finite sums. (a) sin(1) to within 10^-12; (b) e to within 10^-18: [b]3. I know that the taylor expansion for e is e=\sum_{n=1}^{\infty}\frac{1}x^{n}/n! and I aslo know that...
  2. B

    How Do Taylor Series Help Solve Water Wave Velocity Problems?

    Homework Statement A water wave has length L moves with velocity V across body of water with depth d, then v^2=gL/2pi•tanh(2pi•d/L) A) if water is deep, show that v^2~(gL/2pi)^1/2 B) if shallow use maclairin series for tanh to show v~(gd)^1/2 Homework Equations Up above [b]3. The...
  3. R

    Discover P5(x) and 4th Order Taylor Series of Sin(x) and xSin(2x)

    Find P5(x), the 5th order Taylor series, of sin (x) about x = 0. Hence find the 4th order Taylor series for x sin (2x) about x = 0. In this question why is it required to find the 5th order taylor series of sin(x) to find the 4th order taylor series of xsin(2x)?
  4. mnb96

    Geometric intepretation of Taylor series

    Sorry, the title should be: geometric intepretation of moments My question is: does the formula of the moments have a geometrical interpreation? It is defined as: m(p) = \int{x^{p}f(x)dx} If you can't see the formula it is here too: http://en.wikipedia.org/wiki/Moment_(mathematics) with c=0...
  5. J

    How can Taylor series be used to prove a difference involving logarithms?

    Homework Statement Prove if t > 1 then log(t) - \int^{t+1}_{t}log(x) dx differs from -\frac{t}{2} by less than \frac{t^2}{6} Homework Equations Hint: Work out the integral using Taylor series for log(1+x) at the point 0 The Attempt at a Solution Using substitution I get...
  6. N

    Taylor Series for f(x) with nth Derivatives and Coefficients | Homework Help

    Homework Statement Let f be a function with derivatives of all orders and for which f(2)=7. When n is odd, the nth derivative of f at x=2 is 0. When n is even and n=>2, the nth derivative of f at x=2 is given by f(n) (2)= (n-1)!/3n a. Write the sixth-degree Taylor polynomial for f about...
  7. A

    Error Approximation Associated with Taylor Series

    Homework Statement Q1) Use the Taylor series of f (x), centered at x0 to show that F1 =[ f (x + h) - f (x)]/h F2 =[ f (x) - f (x - h) ]/h F3 =[ f (x + h) - f (x - h) ]/2h F4 =[ f (x - 2h) - 8 f (x - h) + 8 f (x + h) - f (x + 2h) ]/12h are all estimates of f '(x). What is the error...
  8. G

    Taylor Series for ln(1-3x) about x = 0 | Homework Question

    Homework Statement Determine the Taylor Series for f(x) = ln(1-3x) about x = 0Homework Equations ln(1+x) = \sum\fract(-1)^n^+^1 x^n /{n}The Attempt at a Solution ln(1-3x) = ln(1+(-3x)) ln(1+(-3x)) = \sum\fract(-1)^n^+^2 x^3^n /{n} Is that right?
  9. K

    Can I Derive the Taylor Series and Radius of Convergence for Tanh(x)?

    Hi. How can I derive the Taylor series expansion and the radius of convergence for hyperbolic tangent tanh(x) around the point x=0. I can find the expression for the above in various sites, but the proof is'nt discussed. I guess the above question reduces to how can I get the expression...
  10. B

    Taylor series of 1/(1+x^2).around x=1

    I know that the Taylor Series of f(x)= \frac{1}{1+x^2} around x0 = 0 is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1 But what I want is to construct the Taylor Series of f(x)=...
  11. B

    An issue with solving an IVP by Taylor Series

    Okay so suppose I have the Initial Value Problem: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{IVP} NB. I am considering only real functions of real variables. If f(x,y) is...
  12. B

    Taylor series of real function with zero radius of convergence

    Can anyone please give me an example of a real function that is indefinitely derivable at some point x=a, and whose Taylor series centered around that point only converges at that point? I've searched and searched but I can't come up with an example:P Thank you:)
  13. T

    Finding the Taylor Series for y(x)=sin^2x

    how to find the taylor series for y(x)=\sin^2 x i need to develop a general series which reaches to the n'th member so i can't keep doing derivatives on this function till the n'th member how to solve this??
  14. E

    Power series vs. taylor series

    Hey all, So I have a physics final coming up and I have been reviewing series. I realized that I'm not quite sure on what the differences are between a Taylor series and a power series. From what I think is true, a taylor series is essentially a specific type of power series. Would it be...
  15. N

    How Accurate Are Partial Sums in Estimating e^N?

    It is known that \sum\limits_{k = 0}^\infty {\frac{{N^k }}{{k!}}} = e^N I am looking for any asymptotic approximation which gives \sum\limits_{k = 0}^M {\frac{{N^k }}{{k!}}} = ? where M\leq N an integer. This is not an homework
  16. G

    Prove periodicity of exp/sin/cos from Taylor series?

    How is it possible to see that exp(i\phi) is periodic with period 2\pi from the Taylor series? So basically it boils down to if is it easy to see that \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}(2\pi)^{2n}=1 ? Or any other suggestions?
  17. R

    This is called the first order approximation or the linear approximation.

    Homework Statement Expand V(z + dz, t). I have seen problems like this in both my EnM and semiconductor courses but it's bothering me because I don't understand how the Taylor series is being used in this case... Homework Equations The Attempt at a Solution Taylor series...
  18. H

    How can Taylor expansion show that the one-sided formula is O(h^2)?

    Homework Statement Using Taylor expansion, show that the one-sided formula (f_-2-4f_-1+3f)/2h is indeed O(h2). Here f-2, for example, stands for f(xo-2h), and f-1 = f(xo-h), so on. The Attempt at a Solution Can some1 help me get starte, I am greatly confused
  19. S

    Finding the Limit: The Taylor Series Approach

    Homework Statement I need to find the following limit. Homework Equations \lim_{x\rightarrow0}\frac{(x-\sinh x)(\cosh x- \cos x)}{(5+\sin x \ln x) \sin^3 x (e^{x^2}-1)} The Attempt at a Solution I think it's got to be something with Taylor series, but I don't really know how to do it.
  20. T

    Taylor Series Tips: Learn & Understand Power Series

    I really need some tips on taylor series...Im trying to learn it myself but i couldn't understand what's on the book... Can anyone who has learned this give me some tips...like what's the difference between it and power series (i know it's one kind of power series), why people develop it, and...
  21. O

    Integration of Taylor Series with Higher Derivatives

    Questions: Is there a quicker way to find the formula for the nth derivative of a function, instead of finding the first several derivatives and trying to find a pattern, and using that pattern to form the equation for the nth derivative? Also, is there a formula for the nth derivative...
  22. S

    Expanding a small oscillation potential in taylor series

    I was wondering if someone could help me with Goldstein's equation 6.3 (3rd Edition). It is the chapter of oscillations and all that he has done in the equation is to expand it in the form of a Taylor series. I can't seem to get how all those ni's come to get there.
  23. A

    Finding the Taylor Series of (1+z)/(1-z) for |z|<1

    Homework Statement Find the taylor series of \frac{1+z}{1-z} where z is a complex number and |z| < 1 Homework Equations \sum^{\infty}_{0} z^n = \frac{1}{1-z} if |z| < 1 The Attempt at a Solution \sum^{\infty}_{0} z^n = \frac{1}{1-z} \frac{1+z}{1-z} =...
  24. M

    Hmmmm how to find the taylor series based @ b for this function?

    Homework Statement 1/(4x-5) - z/(3x-2) based @ 0, answers are in those z things.. sigma Homework Equations i think we use sigma of e^x, but idk how... The Attempt at a Solution since tayor sereis of e^x is like 1/x, do i plug 4x-5 in? thanks
  25. M

    Taylor series just one question pretty easy one thanks(not answer/solution)

    Homework Statement how to you find like the answer for f(1.5), or f(1.00001) those kind of question? thanks with like eq. = f(b)(x-b)... am i making sense? thanks
  26. N

    Calculating Uncertainty in Mass of a Star Using Taylor Series Expansion

    Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...
  27. F

    Not sure I get the Taylor Series

    not sure I get the Taylor Series... Hello Everyone. I understand that the taylor series approximate a function locally about a point, within the radius of convergence. If we use the Taylor series it means that we do not know the function itself. But to find the taylor series we need the...
  28. kreil

    Taylor series expansion of tangent

    Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...
  29. E

    How Do You Expand a Differential Equation Solution into a Taylor Series?

    Many of you have probably used the book Differential Equations by Lomen & Lovelock. For my class I'm working on Example 2, Page 153. You don't need to see the book, though, to help me out. It's a four-part problem and I'm on the last step not knowing where to take it. In Part B, we...
  30. T

    More Taylor series stuff, HELP

    Homework Statement Let T_(4)(x): be the Taylor polynomial of degree 4 of the function ` f(x) = ln(1+x) ` at `a = 0 `. Suppose you approximate ` f(x) ` by ` T_(4)(x) `, find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the...
  31. T

    Could someone help me get my head around this Taylor Series stuff

    Homework Statement The Taylor series for f(x) = ln(sec(x)) at a = 0 is sum_(n=0to infinity) c(sub n) (x)^n. Find the first few coefficients. The Attempt at a Solution I've been trying to figure out where to start by looking it up...I've seen instructions that each coefficient is...
  32. W

    Taylor Series - Range of values

    Homework Statement im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion. i have never come across ths before - any idea?
  33. O

    Taylor series with 3 variables

    Hi am trying to solve this Taylor series with 3 variables but my result is not equal to the solution- So i think i might be wrong expanding the taylor series, or the solution is not correct Homework Statement Find an a approximated value for the function f(x,y,z) = 2x + ( 1 + y) * sin z at the...
  34. B

    Deduce Taylor Series: (2n choose n) x^n Converges to 1/sqrt(1-4x)

    Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...
  35. B

    Upper Bound Error for Maclaurin Polynomial of Sin(x) on the Interval [0,2]

    Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...
  36. O

    Taylor series and quadratic approximation

    Homework Statement use an appropriate local quadratic approximation to approximate the square root of 36.03 Homework Equations not sure The Attempt at a Solution missed a day of class
  37. A

    Compute Taylor Series & Approximate Integral of Exponential Function

    Problem Statement Compute the Taylor Series expansion of f(x) = exp(-x^2) around 0 and use it to find an approximate value of the integral (from 0 to 0.1) of exp(-t^2) dt Solution Part1: First to compute the Taylor Series - I am pretty sure about this step so I will not give details...
  38. S

    A few questions about the Taylor series

    When I tried to learn the Taylor series , I could not comprehend why a infinite series can represent a function Would anyone be kind enough to teach me the Taylor series? thank you:smile: PS. I am 18 , having the high school Math knowledge including Calculus
  39. R

    Taylor Series Homework: Find Series for f(x)=sin x at a=pi/2

    Homework Statement Find the Taylor series for f(x) = sin x centered at a = pi / 2 Homework Equations The Attempt at a Solution Taylor series is a new series for me. I believe the first step is to start taking the derivative of the Taylor series. f(x) = sinx f'(x) =...
  40. J

    Taylor series / 2nd deriv test

    Homework Statement Use the Taylor series about x = a to verify the second derivative test for a max or min. Show if f'(a) = 0 then f''(a) > 0 implies a min point at x = a ... Hint for a min point you must show that f(x) > f(a) for all x near enough to a. Homework Equations The Attempt at a...
  41. F

    Discovering Maclaurin Series for (1 + x)^(-3) with a Taylor Series Approach

    I am trying to find the maclaurin series for f(x) = (1 + x)^(-3) --> what is the best way of doing this--to make a table and look for a trend in f^(n)?
  42. L

    How Do You Calculate the Maclaurin Series for f(x) = 5(x^2)sin(5x)?

    Homework Statement Find the Maclaurin series of the function f(x) = 5(x^2)sin(5x)Homework Equations \sum(Cn*x^n) The Attempt at a Solution I'm supposed to enter in c3-c7 I already know that c4 and c6 are 0 because the derivative is something*sin(0)=0 but for the odd numbered c's I am having...
  43. B

    Taylor Series Expansion of g(z)=1/(z^3) About z0=2

    Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...
  44. C

    Discover the Taylor Series for 3/(z-4i) about -5 | SOLVED

    [SOLVED] Taylor Series Question I have to find the Taylor series of \frac{3}{z-4i} about -5. Therefore, we want the series in powers of z+5. Now, following the textbook it appears that we want to get this in a form that resembles a geometric series so that we can easily express the Taylor...
  45. F

    What are they asking about this Taylor series?

    I'm unclear on what they are asking in this homework problem. Suppose we know a function f(z) is analytic in the finite z plane apart from singularities at z = i and z=-1. Moreover, let f(z) be given by the Taylor series: f(z)=\displaystyle\sum_{j=0}^{\infty}a_{j}z^{j} where aj is...
  46. R

    Taylor Series Approximation Help

    Homework Statement Use the "Three Term" Taylor's approximation to find approximate values y_1 through y_20 with h=.1 for this Initial Value Problem: y'= cosh(4x^2-2y^2) y(0)=14 And write a computer program to do the grunt work approximation Homework Equations The Attempt...
  47. R

    Proving Taylor Series: Maclaurin vs. Taylor

    How does one prove taylor series? Is it proven the same way as Maclaurin's Series(Which i know is a special case of taylor series) f(x)=A_0+A_1x+A_2x^2+A_3x^3+... f(\alpha)=A_0+A_1\alpha+A_2(\alpha)^2+A_3(\alpha)^3+... this kinda doesn't seem like a good way to prove it...as that is how I...
  48. P

    Very very basic taylor series problem

    Homework Statement Consider f(x) = 1 + x + 2x^2+3x^3. Using Taylor series approxomation, approximate f(x) arround x=x0 and x=0 by a linear function Homework Equations The Attempt at a Solution This is the first time that I have seen Taylor series and I am totally lost on how to...
  49. F

    Evaluate the limit of.What is O(x^2) and how can it be used to evaluate a limit?

    Homework Statement I've been asked to: Use the real Taylor series formulae e^{x} = 1 + x + O(x^{2}) cos x = 1 + O(x^{2}) sin x = x(1 + O(x^{2})) where O(x^{2}) means we are omitting terms proportional to power x^{2} (i.e., \lim_{x\rightarrow0} \frac{O(x^{2})}{x^{2}} = C where C is a...
  50. C

    Deriving Taylor Series: Understanding the Step Escaping Me

    I was going through the derivation of the Taylors series in my book (Engineering Mathematics by Jaggi & Mathur), and there was one step that escaped me. They proved that the derivative of f(x+h) is the same wrt h and wrt (x+h). If someone could explain that, Id be really grateful.
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