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. vishal.ng

    A Taylor series expansion of functional

    I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...
  2. doktorwho

    Find the limit using taylor series

    Homework Statement Using the taylor series at point ##(x=0)## also known as the meclaurin series find the limit of the expression: $$L=\lim_{x \rightarrow 0} \frac{1}{x}\left(\frac{1}{x}-\frac{cosx}{sinx}\right)$$ Homework Equations 3. The Attempt at a Solution [/B] ##L=\lim_{x \rightarrow 0}...
  3. C

    I Convergence of Taylor series in a point implies analyticity

    Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
  4. dykuma

    Convert Partial Fractions & Taylor Series: Solving Complex Equations

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  5. karush

    MHB 206.11.3.27 first three nonzero terms of the Taylor series

    $\textsf{a. Find the first three nonzero terms of the Taylor series $a=\frac{3\pi}{4}$}$ \begin{align} \displaystyle f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\ f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\ f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\...
  6. karush

    MHB 206.11.3.39 Find the first four nonzero terms of the Taylor series

    $\tiny{206.11.3.39}$ $\textsf{a. Find the first four nonzero terms of the Taylor series $a=0$}$ \begin{align} \displaystyle f^0(x)&=(1+x)^{-2} &\therefore \ \ f^0(a)&= 1 \\ f^1(x)&=\frac{-2}{(x+1)^3} &\therefore \ \ f^1(a)&= -2 \\ f^2(x)&=\frac{6}{(x+1)^4} &\therefore \ \ f^2(a)&= 6 \\...
  7. Kaura

    Taylor Series Error Integration

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  8. Kaura

    How Can You Simplify the Taylor Series Calculation for cos(3x^2)?

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  9. mastrofoffi

    Derivation of Taylor Series in R^n

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  10. T

    I How Does the -1 Arise in This Series to Function Conversion?

    I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how. $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$ I get most of the function, I just can't see where the ##-1## comes from. Could...
  11. K

    Performing a Taylor Series Expansion for Lorentz Factor

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  12. S

    I Linearizing vectors using Taylor Series

    I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...
  13. T

    Taylor series representation for $$ \frac{x}{(1+4x)^2}$$

    Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...
  14. S

    I Is a function better approximated by a line in some regions?

    I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...
  15. A

    I Taylor Series: What Is the Significance of the a?

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  16. S

    Find Taylor Series for 1/x Around x=3

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  17. Jezza

    How Can the Limit of (ln(1+x))^x as x Approaches 0 Be Evaluated Correctly?

    Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...
  18. A

    I Regarding Error Bound of Taylor Series

    Hi all, I am very confused about how one can find the upper bound for a Taylor series.. I know its general expression, which always tells me to find the (n+1)th derivative of a certain function and use the equation f(n+1)(c) (x-a)n+1/(n+1)! for c belongs to [a,x] However, there are...
  19. bananabandana

    I What Is the Correct Expansion of \( e^{\frac{1}{z-1}} \)?

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  20. A

    MHB Simple to understand derivations similar to the Taylor Series

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  21. deagledoubleg

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  22. nfcfox

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  23. I

    Checking Taylor Series Result of 6x^3-3x^2+4x+5

    Homework Statement Use zero- through third-order Taylor series expansion f(x) = 6x3 − 3x2 + 4x + 5 Using x0=1 and h =1. Once I found that the Taylor Series value is 49. I want to be able to check the value. On the board our teacher plugged in a value into the equation to show that the answer...
  24. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  25. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  26. nomadreid

    Interval of convergence for Taylor series exp of 1/x^2

    Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...
  27. N

    Taylor Series (Derivative question)

    I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x) I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)? Is there some trig identity that I am not aware...
  28. NicolasPan

    Difference between Taylor Series and Taylor Polynomials?

    Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference? Thanks in advance
  29. I

    Finding a taylor series by substitution

    Hello, In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result. For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of...
  30. I

    MHB Is this the correct approach for using Taylor series in this problem?

    Hi there! I need a bit of help on a homework problem. The problem is about a voltage (V) across a circuit with a resistor (R) and and inductor (L). The current at time "t" is: I= (V/R)(1/e^(-RT/L) And the problem asks me to use Taylor series to deduce that I is approximately equal to (Vt/L) if...
  31. Devin

    Taylor Exansion Series Derivation

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  32. N

    Strange invocation of Taylor series

    Hi all, I was working through a chapter on Lagrangians when I cam across this: "Using a Taylor expansion, the potential can be approximated as ## V(x+ \epsilon) \approx V(x)+\epsilon \frac{dV}{dx} ##" Now this looks nothing like any taylor expansion I've seen before. I'm used to ## f(x)...
  33. T

    Expanding a function in terms of a vector

    Homework Statement ## L (v^2 + 2 \pmb{v} \cdot \pmb{ \epsilon } ~ + \pmb{ \epsilon} ^2)##, where ## \pmb{\epsilon}## is infinitesimal and ##\pmb{v}## is a constant vector (## v^2 ## here means ## \pmb{v} \cdot \pmb{v} ## ), must be expanded in terms of powers of ## \pmb{\epsilon} ## to give...
  34. S

    Taylor series for ##\cos^2(x)##

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  35. C

    Summing Taylor Series: Tips & Tricks

    Expanding the series to the n^{th} derivative isn't so hard, however I'm having trouble with the summation. Any tips for the summation? e.g. taylor series for sinx around x=0 in summation notation is \sum^\infty_{n=0} \frac{x^{4n}}{2n!} Thanks.
  36. J

    Proof Taylor series of (1-x)^(-1/2) converges to function

    Hello, I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this: Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n My attempt is to use the lagrange error bound, which is...
  37. P

    Taylor series with using geometric series

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  38. C

    Taylor series expansion of an exponential generates Hermite

    Homework Statement "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Homework Equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by...
  39. M

    Finding Taylor Series for Exponential Functions

    Hello, For the exercises in my textbook the directions state: "Use power series operations to find the Taylor series at x=0 for the functions..." But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n. Here...
  40. C

    Does cos(sqrt(x)) have a valid Taylor series expansion at a=0?

    Find the Taylor series about a=0 for the function F(x) = \cos(\sqrt{x}). Taylor series expansion of a function f(x) about a \sum^{\infty}_0 \frac{f^{(n)}(a)}{n!}(x-a)^n Taylor series of \cos{x} about a=0 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \ldots From these...
  41. A

    Conceptual: Are all MacLaurin Series = to their Power Series?

    Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...
  42. DrPapper

    Exploring Mary Boas' Theorem III: Analytic Functions & Taylor Series

    On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...
  43. END

    How do we define "linear" for single and multivariable?

    In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that xy was listed as a nonlinear term. In...
  44. P

    Differentials, taylor series, and function notation

    "Expanding the taylor series for ##f(x)##.." (See picture) is this a typo? Aren't we expanding ##f(x + \Delta x)##? Also, when we evaluate ##f(x)## (coefficients in the expansion), are we assuming ##\Delta x = 0## by setting ##x + \Delta x## (argument of the function) equal to ##x##? Or are we...
  45. P

    Understanding Taylor Series: Finding the General Formula | Math Explained"

    $$f(a + x) = \sum_{k=0}^∞ \frac{f^{(k)}(a) x^k}{k!}$$ Usually written as: $$f(t) = \sum_{k=0}^∞ \frac{f^{(k)}(a) (t-a)^k}{k!}$$ Where ##t = a + x## Is the taylor expansion supposed to give the same result for all ##a##? The reason this confuses me is because this seems to suggest that ##f(1 +...
  46. M

    Expand a Function with Taylor Series: Quick & Easy

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  47. C

    Statistical Mechanics Mean Field Model

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  48. T

    Estimate number of terms needed for taylor polynomial

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  49. N

    MATLAB Optimizing Taylor Series Approximations in Matlab for Trigonometric Functions

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  50. M

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    I am studying power series right now and I am understanding well how to write them and where they converge but I am having some trouble grasping the Taylor Remainder Theorem for a few reasons. First of all it says the remainder is: f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x. I...
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