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

    Find Help w/ Taylor Series: (y+dy)^0.5

    help with the following taylor series: (y+dy)^0.5 Thanks
  2. eifphysics

    Taylor Series for cos(x^5) | Computing f^(90)(0) | Homework Solution

    Homework Statement Let f(x)=cos(x^5). By considering the Taylor series for f around 0, compute f^(90)(0). by the way, I don't know how super/sub script works? Homework EquationsThe Attempt at a Solution I tried to substitute x^5 into x's Tyler Series form and solve for f^(90)(0), but it gave...
  3. M

    Estimating Remainders for Taylor Series of Sin(x)

    I am just trying to clarify this point which I am unsure about: If I am asked to write out (for example) a third order taylor polynomial for sin(x), does that mean I would write out 3 terms of the series OR to the x^3 term. x-x^3/3!+x^5/5! or just x-x^3/3!Also, I have a question for the...
  4. polygamma

    Convergence of Taylor Series and Definite Integrals: Exploring the Relationship

    Homework Statement If \int_{0}^{1} f(x) g(x) \ dx converges, and assuming g(x) can be expanded in a Taylor series at x=0 that converges to g(x) for |x| < 1 (and perhaps for x= -1 as well), will it always be true that \int_{0}^{1} f(x) g(x) \ dx = \int_{0}^{1} f(x) \sum_{n=0}^{\infty}...
  5. M

    Taylor series generated at x=a

    I have just started learning about series and I don't see the benefit of shifting the series by using some "a" other than 0? My textbook doesn't really tell the benefits it just says "it is very useful"'
  6. A

    Calculating Taylor Series Remainder: Finding an Upper Bound for n

    How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out...
  7. L

    Binomial vs Geometric form for Taylor Series

    Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...
  8. B

    Facing problem in analysing Taylor series expansion

    This is a very basic question . Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it) But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x. Or if asked some other function such as...
  9. ironman

    Does Taylor Series accurately represent limits in calculus?

    Homework Statement [/B] lim x -> 0 2. Homework Equations Taylor series for sin cos e and ln () The Attempt at a Solution I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this: Now the problem is that WolframAlpha says it should be -6/25. Now if...
  10. RJLiberator

    Evaluating the remainder of a Taylor Series Polynomial

    Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...
  11. grandpa2390

    How do I use a taylor series expansion to find effective spring constant

    Homework Statement Let's pretend I am given a potential energy function and nothing else. I need to find the effective spring constant for oscillation about the equilibrium point using a taylor series expansion. I can't find an example or explanation anywhere on how to do this. the potential...
  12. P

    Taylor series to find value of nth derivative

    Homework Statement If f(x) = x^5*cos(x^6) find f40(0) and f41(0) The Attempt at a Solution So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?
  13. T

    Taylor Polynomial of 3rd order in 0 to f(x) = sin(arctan (x))

    The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...
  14. P

    Series Tests and Taylor Series

    Have a quick question about taylor series. We covered taylor series somewhat in class, but there was a complete lack of explanation and our calculus book literally covers the topic in a single page. I understand the idea of a taylor series and how its related to a power series, but what I don't...
  15. T

    Calculating the Taylor Series for cos(x) in Powers of x-pi | Homework Help

    Homework Statement Find the taylor series representation for the following function f(x) = cos(x) in powers of x-pi Homework Equations The Attempt at a Solution [/B] I don't know what they mean by "in powers of x-pi", that's the part I'm confused with. Can somebody please explain that part...
  16. G

    Taylor Series and Random Variables

    Homework Statement A standard procedure for finding an approximate mean and variance of a function of a variable is to use a Taylor Expansion for the function about the mean of the variable. Suppose the variable is y, and that its mean and standard deviation are "u" and "o". f(y) = f(u) +...
  17. C

    Differentiation of a Taylor series

    Homework Statement Hi guys, any help on this question would be hugely appreciated. The Taylor series about 0 for the function f(x)=(1/4+x)-3/2 is f(x)=8 - 48x + 240x^2 - 1120x^3 + ... used differentiation to find the Taylor series about 0 for the function g(x)=(1/4+x)-5/2 The...
  18. I

    MHB Taylor Series Expansion and Radius of Convergence for $f(x)=x^4-3x^2+1$

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  19. E

    Another maclaurin vs. taylor series question

    Hey guys, Struggling with understanding this taylor vs. maclaurin series stuff. So a few questions. Let's say that we have some function f(x). 1. By saying that we want to find the power series of f(x) and nothing else, are we implicitly stating that we are looking for a maclaurin...
  20. A

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  21. B

    Taylor Series Expansion About a Local Minimum

    Hello everyone, I am currently reading chapter two, section 3 of Griffiths Quantum Mechanics textbook. Here is an excerpt that is giving me some difficulty: "Formally, if we expand V(x) in a Taylor series about the minimum: V(x) = V(x_0) + V'(x_0) (x-x_0) + \frac{1}{2} V''(x_0)(x-x_0)^2...
  22. M

    Taylor series representation help

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  23. R

    Understanding Taylor Series Approximation with Taylor's Theorem Explanation

    I'm reading a derivation and it says that the following approximation can be used: I do not under stand how Taylor's theorem allows for this approximation. Can anyone explain this a little?
  24. A

    Where can I learn taylor series and combinatorics?

    I want to learn combinatorics. Please send links? If possible, can you explain now?
  25. B

    Taylor Series coefficient of x^n

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  26. A

    What is the next step for part iii in Taylor Series Extrapolation?

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  27. C

    What is the 13th Taylor coefficient of f(x) at x=3?

    Homework Statement F(x)=7x Determine the 13th taylor coefficient of the taylor series generated by f at x=3 Homework Equations Well, it looks like I just had to take the derivative, but by the time it gets to the 13th derivative, wouldn't the answer just be zero? The Attempt at a...
  28. C

    "How do I compute the Taylor series for cos(7x^2) at x=0?

    1. Homework Statement [/b] Determine the Taylor series for the function below at x=0 by computing P 5 (x) f(x)=cos(7x^2) Homework Equations I used to taylor series for cosx and replaced it with 7x^2 so i used 1-49x^4/2! +2401x^8/4!... and so on. That should be correct, my attempt...
  29. A

    Estimating Bandwidth of Phase Modulated Signal Using Taylor Series

    Homework Statement Consider the PM (phase modulated) signal, s(t) = Acos(wt+x(t)) where x(t) is the information bearing signal. Assume that |x(t)|< y, which is not necessarily small. Using Taylor's series expansion, derive an estimate for the bandwidth of the PM signal s(t). Homework...
  30. A

    MHB Taylor Series Applications

    What is the general procedure for using Taylor Series to evaluate: i) sums eg.\sum_{n=4}^{\infty }\frac{n(n-1)2^n}{3^n} ii) limits eg. \lim_{x\rightarrow 2}\frac{x^2-4}{ln(x-1)} iii) derivatives eg. Find f^{(11)}(0) of f(x)=x^3sin(x^2) iv) integrals eg. \int_{0}^{1} \frac{1}{2-x^3}dx
  31. M

    MHB Linear approximation of Non linear system by Taylor series

    I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...
  32. A

    MHB Taylor Series: Find 2nd Degree Series for x^2+y^2=4 at [1, -\sqrt[]{3}]

    How would I find the second-degree Taylor series for x^2+y^2=4 at [1, -\sqrt[]{3}]?
  33. J

    Struggling with this limit value.(probably using taylor series)

    Struggling with this limit value Homework Statement Calculate lim((e^x-1)/x)^(1/sin(x)) where x\rightarrow0 Homework Equations Maclaurin series. sin(x)/x -----> 1 when x->0 (possibly) The Attempt at a Solution (e^x-1)/x)^(1/sin(x) = ((x+x^2/2+x^3H(x))/x)^(1/sin(x))...
  34. J

    Taylor series in terms of discrete derivative

    All analitic function can be express how: f(x) = \frac{1}{0!} \frac{d^0f}{dx^0}(x_0) (x - x_0)^0 + \frac{1}{1!} \frac{d^1 f}{dx^1}(x_0) (x - x_0)^1 + \frac{1}{2!} \frac{d^2f}{dx^2}(x_0) (x - x_0)^2 + \frac{1}{3!} \frac{d^3f}{dx^3}(x_0) (x - x_0)^3 + ... that is the taylor series of the function...
  35. L

    Taylor series problem (non-direct differentiation?)

    I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?
  36. JasonHathaway

    Taylor series with two variables

    Hi everyone, Homework Statement + Homework Equations + The Attempt at a Solution
  37. F

    MHB Taylor Series Expansion of $e^{-(q+s)^2}$

    Use taylor series to show that the infinite series from n=0 of $$\frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})=e^{-(q+s)^2}$$
  38. L

    Complex Analysis - Taylor series of 1/(1+exp(z))

    Homework Statement Compute the first four terms of the Taylor series of \frac{1}{1+e^{z}} at z_{0} = 0 and give it's radius of convergence. Homework Equations e^{z} = \sum\frac{z^{n}}{n!} = 1 + z +\frac{z^{2}}{2!} + \frac{z^{3}}{3!} + o(z^{3}) \frac{1}{1+w} =...
  39. I

    Taylor Series, Binomial Series, Third Order Optics

    Homework Statement Show that if cosΦ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for ℓ^{-1}_o and ℓ^{-1}_i. Also, use Φ ≈ sinΦ.] Homework Equations Sorry that...
  40. X

    Taylor Series Help: Determine Real Number

    Homework Statement Determine the real number to which the series \sum^{∞}_{k=1} (2-e)^k/2^k(k!) Homework Equations I know that e^x = the series of x^k / k The Attempt at a Solution I would assume to sub in 2-e for x, but then that takes away the x.
  41. X

    Use Taylor Series To Evaluate

    1. Homework Statement [/b] use taylor series to evaluate lim x -> 0 of \frac{ln(x)}{(x-1)}[b] Homework Equations I know that -ln (1-x) taylor polynomial and that of ln (1+x) The Attempt at a Solution Using the basics that I know I would assume I would just make ln (1+x) = ln (x)...
  42. J

    Taylor series for a complex function

    Homework Statement Find the 5 jet of the following function at z=0: f(z) = \frac{sinhz}{1+exp(z^3)} Homework Equations \frac{1}{1-z}=\sum_{n=0}^\infty z^n where z=-exp(z^3) The Attempt at a Solution I have tried to multiply the series for sinhz by the series for \frac{1}{1-(-exp(z^3))} but...
  43. H

    Where Can I Find Information on Testing the Convergence of Taylor Series?

    Homework Statement Where does the Taylor series converge? [You do not need to find the Taylor Series itself] f(x)=... I have a few of these, so I'm mainly curious about how to do this in general. The Attempt at a Solution I haven't really made an attempt yet. If I were to make an...
  44. Feodalherren

    Taylor series integration of cosx -1 / x

    Homework Statement ∫((cosX)-1)/x dx Homework Equations Taylor Series The Attempt at a Solution My approach was basically to to split the integral into two more manageable parts which gave me ∫(cosX/x)dx - ∫(1/x)dx The solutions manual did it completely differently and...
  45. E

    Taylor Series for Complex Variables

    Homework Statement Obtain the Taylor series ez=e Ʃ(z-1)n/n! for 0\leq(n)<\infty, (|z-1|<\infty) for the function f(z)=ez by (ii) writing ez=ez-1e. Homework Equations Taylor series: f(z) = Ʃ(1/2\pi/i ∫(f(z)/(z-z0)n+1dz)(z-z0)n The Attempt at a Solution The first part of this...
  46. V

    How Do I Transform Coefficients in a Taylor Series Differential Equation?

    So, I have this DE which is 2nd order, w/ variable coefficients, it goes; xy''+(x-5)y'+(x^2-4)y=0 revolving around x_0=4. I know there's a singular point at 0 and I assume to use a summation y(x)=[∞,Ʃ,n=0] a_n(x-x_0)^n pardon me I don't know how to type the summation symbol, but that's...
  47. Z

    JasonWhat is the Taylor series for ln(x+2) about x = 0?

    Homework Statement Using power series, expand ln(x + 2) about a = 0 (Taylor series) Homework EquationsThe Attempt at a Solution Is this appropriate? ln(x+2) = ln((x+1)+1) x' = x+1 ln(x'+1) = \sum_{n=0}^{\inf} \frac{(-1)^n}{n+1}(x')^{n+1} or ln(x+2) = ln(\frac{x}{2}+1) x' =...
  48. M

    Expanding Inhomogeneous Poisson Processes Using Taylor Series

    I'm at the end of a very long Poisson Processes question, involving inhomogeneous Poisson Processes. I just need to be able to expand the following expression to be able to complete the question. exp[{(sin ∏h)/∏} -h] Would anyone please be able to provide some help, with steps please!
  49. TheFerruccio

    How do I find the Taylor Series of ##\frac{q}{\sqrt{1+x}}## around x = 0?

    This is rather embarrassing, because I should have known how to do this for years. Question: Compute the Taylor Series of ##\frac{q}{\sqrt{1+x}}## about x = 0. Attempt at Solution: Term-wise, I have gotten... ##f(0)+f'(0)+f''(0)+... =...
  50. C

    The resulting webpage title could be: Simplifying Limits with Taylor Series

    Homework Statement \lim_{x \to 0}[\frac{\sin(\tan(x))-\tan(\sin(x))}{x^7}]Homework Equations \sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} + ... \tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{215}+ ...The Attempt at a Solution I have an idea of how to do this by replacing...
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