Maclaurin series Definition and 155 Threads

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. Y

    Need help with maclaurin series question

    its using the Maclaurin series, i have already worked out the equations: cos x = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + (x^8)/8! - (x^10)/10! + ... e^x = 1 + x + (x^2)/2! +(x^3)/3! + (x^4)/4! + (x^5)/5! + (x^6)/6! + ... how do i use these two results to obtain the first 6 terms on the...
  2. W

    Can someone help me understand Taylor and MacLaurin series?

    I am having difficulty understanding Taylor and MacLaurin series. I need someone to go through step by step and explain a problem from beginning to end. You could use the function f(x) = cos x. Also, could someone find the MacLaurin series of 1/(x^2 + 4) ? I just don't understand the basics of...
  3. P

    Find 1st 4 Non-Zero Terms Maclaurin Series

    Im helping my sis study for her exam but i can't remember how to find the first four non-zero terms of the maclaurin series
  4. D

    CLUELESS about a Maclaurin series

    CLUELESS about a Maclaurin series! I'm supposed to obtain a Maclaurin series for the function defined by f(x) = \left\{ \begin{array}{lc} e^{-1/x^2} & \mbox{ if } x \neq 0 \\ 0 & \mbox{ if } x = 0 \end{array} \right. I get immediately stuck as I find: f(0) = 0 f^{\prime}(0) =...
  5. M

    Maclaurin Series and the general term

    using the sin and cos Maclaurin series, validate each of them using at least 3 values for x and determine how many terms are needed to provide reasonable accuracy. Find the General Term (Tn where n = 1, 2, 3, ...) for each expression and show that each correctly generates the terms of the...
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