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. I

    Taylor Series: Equivalence of Two Forms Explained

    I don't get how these two forms of the taylor series are equivalent: f(x+h)= \sum_{k=0}^{\infty} \frac{f^k(x)}{k!} h^k f(x) = \sum_{k=0}^{\infty} \frac{f^k(0)}{k!}x^k The second one makes sense but I just can't derive the first form using the second. I know its something very simple...
  2. T

    Taylor Series for exp(x^3) around x = 2

    Homework Statement Give the Taylor Series for exp(x^3) around x = 2. Homework Equations f(x) = Sum[f(nth derivative)(x-2)^n]/n! The Attempt at a Solution I know the solution for e^x but can't seem to find a formula for the nth derivative of exp(x^3) around x = 2. Thanks for...
  3. T

    Taylor Series Help: ƒ(x) = e^(x/2), g(x) = ((e^(x/2)) - 1)/x

    Let ƒ be the function given by f (x) = e ^ (x / 2) (a) Write the first four nonzero terms and the general term for the Taylor series expansion of ƒ(x) about x = 0. (b) Use the result from part (a) to write the first three nonzero terms and the general term of the series expansion about x = 0...
  4. G

    What is the Linear Approximation Range for Sin(f) Within 10% Error?

    Homework Statement For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 % Homework Equations The taylor series from 0 to 1 , the linearization, is the most appropriate equation The Attempt at a Solution...
  5. D

    Complex Taylor series

    TASK: Assuming a complex function f(z) can be expanded as a Taylor series around z=0, i.e.: f(z)=\sum_{n=0}^{\infty}a_{n}z^n Setting z=r*exp(i*theta), assuming a_n is real, find real part u(r, theta), imaginary part v(r,theta). Comment the result, especially for r=1. MY SOLUTION...
  6. K

    Taylor Polynomial (can you help me?)

    1) Let f(x) = (x^3) [cos(x^2)]. a) Find P_(4n+3) (x) (the 4n + 3-rd Taylor polynomial of f(x) ) b) Find f^(n) (0) for all natural numbers n. (the n-th derivative of f evaluated at 0) I know the definition of Taylor polynomial but I am still unable to do this quesiton. I tried to find the...
  7. G

    Linearizing a system using a taylor expansion

    Homework Statement Linearize the system operator illustrated below by applying a Taylor series expansion. f(t) ----> e^f(t) -----> g(t) Homework Equations I only find the general form of a taylor series relevant. g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n! The system is...
  8. M

    Calculate nth Degree Taylor Polynomial for f(x)=sqrtx | Taylor Polynomial Help

    Homework Statement I'm trying to make the nth degree taylor polynomial for f(x)=sqrtx centered at 4 and then approximate sqrt(4.1) using the 5th degree polynomial I know that the polynomials are found using the form: P(x)= f(x)+f'(x)x+f''(x)x^2/2factorial...f^n(x)x^n/nfactorial so...
  9. P

    Taylor Expansion Without Variables?

    This is just part of a larger problem, but I have a basic equation r'=k-g*r, where k and a start out as constants, but then I need to treat everything as if it can vary slightly from the average. For this, I set r=r_ave+dr, g=g_ave+dg, and k=k_ave+dk. Now I need to work these into the first...
  10. S

    Taylor polynomials for multivariable functions

    Ok there's something I don't get. I know for instance that the linear polynomial for say f = 91 + 2x + 3y + 8z + Quadratic(x, y, z) + Cubic(x, y, z) ... is 91 + 2x + 3y + 8z if the base point is (0, 0, 0). This is pretty clear. What I don't get is why when you take the base point to be say (1...
  11. T

    Can the Taylor Series of Analytic Functions be Proven?

    i'm having a hard time understanding taylor series and why it works and how it works. if someone could please explain it to me that would be great. My teacher explained it in class but he goes so fast that i have no idea what he's saying. he did give us some practice problems but if i have no...
  12. P

    Taylor Expansion for ln(1+x)/(1-x) About x=0

    Hi I wonder if there is a simpler way to obtain the first three non-zero terms of Taylor Expansion for the function \frac{Ln(1+x)}{1-x} about x=0? I differentiated it directly, but it was such a nightmare to do:mad: . So I am wondering if there is a simpler way to do it?
  13. M

    Understanding the Taylor Series Concept: Exploring its Uses and Applications

    hi everyone, I am just learning the taylor series at school. I am slightly confused. in my textbook, one of hte exercises is to find hte nth degree taylor polynomial of x^4 about a=-1. n is 4 in this case so this gives me a long polynomial. i understand that inputting any x value into this...
  14. K

    Deriving Quadratic Equation from Taylor Expansion: An Exercise

    Homework Statement I have the following question to answer: Show that (X^2/h^2)*((1/2*y1) - y2 + (1/2*y3)) + (X/h)*((-1/2 y1)+(1/2 y3))+y2 (sorry about the format) is equal to (taylor expansion): y = y2+(x(dy/dx)¦0 + (x^2/2*((d^2)y)/(dx^2))¦0 Homework Equations also given in...
  15. C

    3rd order, multi variable taylor polynomial

    any insight to this question? .. i mean.. usually people just do up to order 2.. find the taylor polynomial of order 3 based at (x, y) = (0, 0) for the function f(x, y) = (e^(x-2y)) / (1 + x^2 - y) how large do you have to take k so that the kth order taylor polynomial f about (0, 0)...
  16. V

    How Does Taylor Series Approximation Determine Electric Potential Over Distance?

    The electric potential V at a distance R along the axis perpendicular to the center of a charged disc with radius a and constant charge density d is give by V = 2pi*d*(SQRT(R^2 +a^2) - R) Show that for large R V = pi*a^2*d / R This is what I have done so far... V = 2pi*d *...
  17. T

    How Can Taylor Series Approximate Second Derivatives?

    I am supposed to prove using taylor series the following: \frac{d^2\Psi}{dx^2} \approx \frac{1}{h^2}[\Psi (x+h) - 2\Psi(x) + \Psi (x-h)] where x is the point where the derivative is evaluated and h is a small quantity. what i have done is used: f(x+h)= f(x) + f'(x) h +...
  18. B

    Taylor Expansion for Gravitational Acceleration Problem

    Hey Everyone. I'm ALMOST finished this problem... To spare you the long story, I need to take the difference between an gravitational acceleration, and the same gravitational acceleration at a slightly larger height. The two functions are a(r) and a(r+d), where d is very small Now... VERY...
  19. K

    Expanding f(x,y) with Double Taylor Series

    Let be an analytic function f(x,y) so we want to take its Taylor series, my question is if we can do this: -First we expand f(x,y) on powers of y considering x a constant so: f(x,y)= \sum_{n=0}^{\infty}a_{n} (x)y^{n} and then we expand a(n,x) for every n into powers of x so we have...
  20. C

    Finding the 2005th Derivative with Taylor Series for Inverse Tan Function

    I have got a question here that puzzles me. How do I use TAYLOR SERIES to find the 2005th derivative for the function when x=0 for the following function: f(x) = inverse tan [(1+x)/(1-x)] Part (1) I was hinted that differentiating inverse tan x is = 1/(1+x^2). Part (2) After which, I need to...
  21. P

    Multivariable Taylor polynomials?

    In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about...
  22. M

    Taylor Polynomial for Square Root Function at x = 100

    Hi Guys, I have an assigment which I would very much appreciate if You would tell if I have done it correct :) Use the Taylor Polynomial for f(x) = \sqrt(x) of degree 2 in x = 100. To the the approximation for the value \sqrt(99) First I find the Taylor polynomial of degree 2...
  23. C

    Taylor Series Approximation for Solving Initial Value Problems

    With a simple ODE like \frac{ds}{dt} = 10 - 9.8t and you're given an initial condition of s(0) = 1, when doing the approximation would s'(0) = 10 - 9.8(0), s'' = ... etc?
  24. A

    Starting a Taylor series problem, .

    Compute the Taylor series for f(x)= sq root (x) about x=1. Determine where the series sconverges absolutely, converges conditionally, and diverges. Hint: 2(k!)=2*4*6...(2k-2)*2k. Also 1<2, 3<4, 5<6,..., 2k-1<2k should help you out with a comparision.
  25. M

    Is This the Correct Taylor Polynomial for sqrt(x) at x=100?

    Hi Given a function f(x) = sqrt(x) is the Taylor Polynomial of degree 2 for that function: \frac{x^2}{2} - 99x + 4901 where x = 100 ? Sincerely Fred
  26. E

    Complex analysis taylor series Q

    hi, I'm wondering if someone can help me out with this question: "What are the first two non-zero terms of the Taylor series f(z) = \frac {sin(z)} {1 - z^4} expanded about z = 0. (Don't use any differentiation. Just cross multiply and do mental arithmetic)" I know the formula for...
  27. S

    Complex Variable Taylor Expansion at z=2i

    I'm having trouble determining the order of the pole of [exp(iz) - 1]/((z^2) + 4) at z=2i I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ... because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does...
  28. G

    Numerical Analysis: Taylor Polynomials, Error, Bounds

    (a) I found the answer to be: 1/(1-x) = 1 + x + x^2 + x^3 + ... + [x^(n+1)]/(1-x) for x != 1 *Note: "^" precedes a superscript, "!=" means "does not equal" (b) Use part (a) to find a Taylor polynomial of a general (3n)th degree for: f(x) = (1/x)*Integral[(1/(1 + t^3), t, 0, x] *Note...
  29. W

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

    the problem reads develop expansion of ln(1+z) of course I just tried throwing it into the formula for taylor expansions, however I do not know what F(a) is, the problem doesn't specify, so how can I use a taylor series?
  30. T

    Taylor Series for ln(1-x): Get Help Now

    may i know from ln(1-x) how to become - [infinity (sum) k=1] x^k / k ? pls help
  31. P

    Spacetime Physics by J. Wheeler and E. Taylor

    I have had this book for a while and never really looked into it. It claims to be an easy/nonmathematical approach to relativity. Has anyone read this book before? Can I really understand what the subject matter is covering without any post-calculus math? Is it also a good beginer's guide to...
  32. P

    Taylor series homework problem

    Dear friends, I have a question on a taylor series, that is this one: A·e^(i (x)) That is: cos (x)+ i sin (x) becouse of the taylor's. But, is this wrong? A·e^(v (x)) = cos (x)+ v sin (x) (v is a vector). Tks.
  33. A

    Calculating Limits with Taylor Series: Tips and Guidelines for Accurate Results

    Taylor rule of thumb?? When calculating limits by using taylor series is there any easy way to know how many elements that should be included in the taylor series? if I have \lim_{x\rightarrow\zero} \frac{exp(x-x^2)-Cos2x-Ln(1+x+2x^2)}{x^3} How do I know many terms to include in...
  34. A

    Taylor Expansion Hints: Find First Non-Zero Term of x = 0

    Can anyone please give me a hint on any of the following Taylor expansions? That would be so helpful! for all three: Find the first non-zero term in the Taylor series about x = 0 problem 1 \frac{1} {sin^2x} - \frac{1} {x^2} everytime I differentiate the result is zero...so that...
  35. A

    Taylor Series to x places of decimals

    Hi all, here's the problem: given: tan^(-1)= x - x^3/3 + x^5/5 using the result tan^(-1) (1)= pi/4 how many terms of the series are needed to calculate pi to ten places of decimals? note: this is supposed to say tan^(-1) and tan^(-1)[1] respectively Does anyone know whether...
  36. N

    What did I do wrong in my Taylor Series Expansion for y=kcosh(x/k)?

    y=kcosh(x/k) = k(e^(x/w) + e^(-x/k)) y ~ k[1 + x/k + (1/2!)(x^2/k^2) + (1/3!)(x^3/k^3)] +...+ k[1 - x/k +(1/2!)(x^2/k^2) - (1/3!)(x^3/k^3) +...] All odd terms except 1 cancel out. So we are left with y = k [2 + (2/2!)(x^2/k^2) + (2/4!)(x^4/k^4) + (2/6!)(x^6/k^6) +...] I've been...
  37. kreil

    Studying Taylor and Maclaurin Series

    So I'm studying Taylor Series (I work ahead of my calc class so that when we cover topics I already know them and they are easier to study..) and tonight I found a formula for taylor series and maclaurin series, and i used them to prove eulers identity. However, I don't really know much about...
  38. N

    Can the Taylor series be inverted without using Lagrange's theorem?

    How can you invert a Taylor serie? x=y+Ay^2+By^3+Cy^4... to y=ax+bx^2+cx^3 ... without the lagrange theorem... must go from x=y+Ay^2+By^3+Cy^4... to y=ax+bx^2+cx^3 ... Need help thanks!
  39. A

    Taylor series of 1/sqrt(cosx)

    Is there a way to get the Taylor series of 1/sqrt(cosx), without using the direct f(x)=f(0)+xf'(0)+(x^2/2!)f''(0)+(x^3/3!)f'''(0)... form, just by manipulating it if you already know the series for cosx?
  40. S

    Taylor Polynomial (n=4) for g(x): Showing 0<E4(x)<80x^5

    Hi I have found the following TP (n=4) for g(x) = (1+5x)^1/5 P4(x) = 1+x-2x^2+6x^3-21x^4 Then they ask me to show that 0<E4(x)<80x^5 when x>0. I don't know how to start, or exactly what I am supposed to show...? I have found E4(x) to be( 399/[5(1+5X)^24/5] ) *x^5... And 0<X<x ...?
  41. P

    Solving Differential Equations with Simple Taylor Series Method

    Hi, I was reading this math book once... and it had a method for solving differential equations of 1st (And maybe 2nd? I don't remember) order by using simple Taylor series... I didn't even have to understand much of what was going on, except that I followed some simple rule and I ended up...
  42. S

    Taylor Polynomial for f(x) = √(x+1) | Approximate & Find Error

    Find the thrid taylor polynomial P3(x) for the function f(x) = \sqrt{x+1} about a=0. Approximate f(0.5) using P3(x) and find actual error thus Maclaurin series f(x) = f(0) + f'(0)x + \frac{f''(0)}{2} x^2 + \frac{f^{3}(0)}{6} x^3 f(x) = x + \frac{1}{2} x - \frac{1}{8} x^2 +...
  43. T

    Approximate the integral int (1 - cos x)/x dx using Taylor expansion

    I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x: \int \frac{1 - cos x}{x}dx and given cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}... i should get a simple series similar to this for...
  44. B

    Understanding Taylor Series Error & Degrees of Variables

    Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...
  45. B

    Exploring the Relationship between Taylor Series Degree & Error

    Hi, can someone explain me the relation between the degree of a taylor series (TS) and the error. It is for my class of numerical method, and I do not find a response to my question in my textbook. I mean when we have a function Q with two variables x and y,and we use a version of TS to...
  46. B

    Solving Taylor Series Problem with m-th Derivative Bound

    Hi , I have some difficulties to solve this problem. It is from my numerical methods class but the problem is about taylor series: It is known that for 4 < x < 6, the absolute value of the m-th derivative of a certain function f(x) is bounded by m times the absolute value of the quadratic...
  47. C

    What is the proof of the Taylor Theorem in n variables?

    Hello, guys. I am studying the Taylor Theorem for functions of n variables and in one book I've found a proof based on the lemma that I am copying here. I am having some trouble in following its proof so I seek your kind assistance. The lemma rests on two items: the definition of a function...
  48. C

    What is the Taylor expansion for 1/(1-exp(-1))?

    Hi How do you expand (1-exp(-1))^-1 as Taylor series Callisto
  49. D

    Taylor differentition polynomials?

    taylor differentition polynomials? hi got a question here that involves this extremely difficult question anyone that can point me in the right direction on what to do will be most appreciated :) Find Exactly the tayor polynomial of degree 4 f(x) = cos ( pi*x / 6 ) about x=-1 i know...
  50. M

    Taylor series expansion question

    Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...
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