What is Taylor: Definition and 873 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. 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)=...
  2. 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...
  3. 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:)
  4. S

    What is the Taylor expansion of ln(1+z)?

    Homework Statement Develop the Taylor expansion of ln(1+z). Homework Equations Taylor Expansion: f(z) = sum (n=0 to infinity) (z-z0)n{f(n)(z0)}/{n!} Cauchy Integral Formula: f(z) = (1/(2*pi*i)) <<Closed Integral>> {dz' f(z')} / {z'-z} The Attempt at a Solution I have NO idea...
  5. T

    Possible webpage title: Proving the Taylor Inequality for Positive Values of x

    prove this inequality for x>0 x-\frac{x^3}{6}+\frac{x^5}{120}>\sin x this is a tailor series for sin x sinx=x-\frac{x^3}{6}+\frac{x^5}{120}+R_5 for this innequality to be correct the remainder must be negative but i can't prove it because there are values for c when the -sin c...
  6. A

    Could someone please explain this (simple) fact about Taylor expansions?

    My professor just told me that if \Delta x is small, then we can expand L(x+\Delta x) as follows: L(x + \Delta x) = L(x) + \frac{d L}{d x} \Delta x + \frac{1}{2!} \frac{d^2 L}{d x^2} (\Delta x)^2 + \ldots, where each of the derivatives above is evaluated at x. Could someone please...
  7. 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??
  8. A

    Taylor Development: Combining cos(z) & cosh(z) in Complex Field

    Homework Statement Hey guys. I need to develop Taylor series for this function (cos(z) * cosh(z)). I know the Taylor development for cos and the Taylor development for cosh but I have no idea how to combine the two, if it's possible, any idea guys? And another thing, does it matters if we...
  9. T

    Finding the first-order taylor polynomial

    Homework Statement Basically, I have a differential equation. One of the elements of it is... F(P) = 0.2P(1 - (P/10)) And I need to replace it with it's first-order Taylor polynomial centered at P=10. The Attempt at a Solution I haven't done Taylor polynomial stuff in over a...
  10. W

    Mathematicians' Original Work: Riemann & Taylor Theorems

    Can anyone provide me with a website that has copies of the original works of Riemann, Taylor, famous mathematicians. I am looking for papers on proved theorems.
  11. 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...
  12. 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
  13. K

    Proving Inequalities of Euler-Mascheron Constant with Taylor Expansion

    Homework Statement With n>1, show that (a) \frac{1}{n}-ln\frac{n}{n-1}<0 and (b) \frac{1}{n}-ln\frac{n+1}{n}>0 Use these inequalities to show that the Euler-Mascheron constant (eq. 5.28 - page330) is finite. Homework Equations This is in the chapter on infinite series, in the section...
  14. F

    Taylor Representation of the Floor Function

    Hi Guys, I was wondering if it is possible (why or why not) to define the floor function, Floor[x], as an infinite Taylor Series centered around x=a? Any sort of help is greatly appreciated! flouran
  15. 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?
  16. 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...
  17. 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
  18. 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.
  19. E

    Taylor Polynomial for f(x)=sec(x)

    Hey all, so I need to find 4th degree taylor polynomial of f(x)=sec(x) centered at c=0 Can I just use substitution to find the answer since sec(x) = 1/cos(x) and I know the taylor series for cos(x). I guess, essentially, can I take the reciprocal of the taylor series of cosx to get sec(x)...
  20. E

    Taylor Polynomial for f(x)=sec(x)

    Hey all, so I need to find 4th degree taylor polynomial of f(x)=sec(x) centered at c=0 Can I just use substitution to find the answer since sec(x) = 1/cos(x) and I know the taylor series for cos(x). I guess, essentially, can I take the reciprocal of the taylor series of cosx to get sec(x)...
  21. 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...
  22. B

    How Does Taylor's Method Apply to Initial Value Problems?

    Consider the IVP: \left. \begin{array}{l} \frac {dy} {dx} = f(x,y) \\ y( x_{0} ) = y_{0} \end{array} \right\} \mbox{ze IVP :p} Hypothesis: f(x,y)\subset C^\infty_{x,y}(D)\; \; / \; \;(x_0,y_0)\in D [Note that this condition automatically satisfies the hypotheses of the...
  23. 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...
  24. 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.
  25. 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} =...
  26. 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
  27. M

    How to find the taylor for sin(x)^2 w/ sin(x), is this right?

    Homework Statement sin(x)= sum((-1)^k* (x^(2k+1)/(2k+1)!))k=0 to infinity Homework Equations so if i want to find sin(x)^2, (not sin(x^2), that would be easier though...) The Attempt at a Solution then... do i square the whole thing, like this? sum(((-1)^k*...
  28. 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
  29. 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...
  30. 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...
  31. C

    Quadratic Approximation of Potential Function using Taylor Expansion Method

    Homework Statement What is the quadratic approximation to the potential function? Homework Equations U(x) = U0((a/x)+(x/a)) U0= 20 a=4 The Attempt at a Solution This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even...
  32. 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...
  33. V

    Taylor Polynomial Homework: Estimating x Range with Error < 0.01

    Homework Statement I can either use the alternating series estimation thereom (which i don't really know) or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. sin(x) = x - (x^3)/6 (|error| < 0.01) Do I...
  34. V

    What is the Taylor Polynomial for Arcsin x at a = 0 and n = 3?

    Homework Statement Find the Taylor polynomial T_n(x) for the function arcsin x at a = 0, n = 3 Homework Equations Well, I understand the Taylor poly. for sine, but how do i get arcsine?
  35. M

    Cauchy Riemann & Taylor Expansion.

    Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...
  36. D

    Using Taylor Formula to Find Series of f(x)=e^{2x}

    Homework Statement using the Taylor Formula, find the series for the function f(x)=e^{2x}Homework Equations \sum \frac{f^{n}(a)}{n!} (x-a)^{n} any help as to where i start would be great. new to series...
  37. B

    Simple Taylor or Multipole Expansion of Potential

    Hullo, Somehow, I couldn't get the TeX to come out right. I have been trying to learn scheme theory (algebraic geometry) and completely forgotten how to do this simple calculus type stuff... Homework Statement Let V be a potential of the form [tex]V = \left(\frac{1}{r} +...
  38. 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...
  39. 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...
  40. 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...
  41. J

    How Do You Solve a Taylor Series Problem with a Differential Equation?

    This is for revision purposes (not homework so I am not trying to cheat my way out of it!) and its too late in the week to see my lecturer about this. I don't have much of an attempt at the solution because i haven't got a clue where to start. It looks like just a short one though. Here goes...
  42. 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?
  43. C

    Integration and taylor expansion

    can anybody help me with this integration? Integral of e to the -2x times x squared dx it expands to 1/4, but I'm not sure how to start.
  44. M

    Finding Taylor Polynomial of Degree 4 for f(x)=sqrt(x) About a=4

    I need to find the Taylor polynomial of degree 4 expanded about a=4 for the function f(x)=squareroot of (x)=x^(1/2) This is what I've started with but I'm not sure how to proceed and if I even started correctly: f'(x)(-1/2)x^(-1/2)=1/2sqrt(x) f"(x)=(-1/4)x^(-3/2)=-1/4x^3/2...
  45. G

    Taylor Polynomial of Order 3 for f(x,y,z) at (0,0,0)

    Homework Statement Calculate the taylor polynom of order 3 at (0,0,0) of the function with well-known series (that means I can't just take the derivatives) f(x,y,z)=\sqrt{e^{-x}+\sin y+z^{2}} Homework Equations The Attempt at a Solution I wrote the functions within the square...
  46. F

    Error in Taylor polynomial of e^x

    Find the Taylor polynomial of degree 9 of f(x) = e^x about x=0 and hence approximate the value of e. Estimate the error in the approximation. I have written the taylor polynomial and evaluated for x=1 to give an approximation of e. Its just the error that is confusing me. I have: R_n(x) =...
  47. B

    How Many Terms Needed in Maclaurin Polynomial for Error Below 0.001?

    Taylor Polynomial Error--Please help! Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 So is the procedure to take the derivatives and plug in 0 (since c=0) and find an...
  48. B

    Taylor polynomial approximation- Help

    Use Taylor's theorem to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function to be less than .001. e^.3 I really, really don't know what to do for this one, and I have a quiz tomorrow. I have read through the section in the book, but...
  49. B

    Really - Taylor Polynomial Approximation Error

    Homework Statement Use Taylor's theorem to obtain an upper bound of the error of the approximation. Then calculate the exact value of the error. cos(.3) is approximately equal to 1 - (.3)^2/2! + (.3)^4/4! Homework Equations The Attempt at a Solution I came up with upper...
  50. C

    How Do You Calculate a Degree 3 Taylor Polynomial for e^x?

    Hello, I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :) Find the Taylor polynomial of degree 3 of f (x) = e^x about x = 0 and hence find an approximate value for...
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