Taylor's and Maclaurin's Series | Research & Encyclopedia Articles

Taylor's and Maclaurin's Series

A Taylor's series is a series expansion that acts as a representation of a function. A series expansion is a representation of a function as a sum of powers in one of its variables or a sum of powers of another function. A more specific form of a Taylor's series is the Maclaurin's series. The Taylor's series is an expansion about an arbitrary point, x = a, whereas a Maclaurin's series is an expansion about zero in particular, x = 0. The main advantage of using a power series representation of a function is that the value of the function at any point is equal to a convergent series and so can be approximated by its partial sums. These power series contributed greatly to the growth of calculus. They allowed mathematicians to analyze properties of functions with a single theory and to approximate values of functions easily. Taylor's series can be used to estimate values for e, ln 2, and .

The Taylor's series of a function f is written as _{n = 0} [f⁽ⁿ⁾(0)/n!] xⁿ. For x = a: f(x) = f(a) + f'(a)(x - a) + ... [f⁽ⁿ⁾(a)/n!](x - a)ⁿ + r_n(x), where r_n(x) is the Lagrange remainder formula and is r_n(x) = [f^{(x + 1)}(t_x)/(n + 1)!](x - a)^{(n + 1)}.

The Lagrange remainder is sometimes called the error after n terms in the Taylor's series. There are power series expansions for many functions; but if a function is to have a power series expansion, then it must have derivatives of all orders on the defined interval of interest. Because a Taylor's series is a power series, it has a circle of convergence whose radius extends to the nearest singularity.

Brook Taylor, an English mathematician, invented the method for expanding functions in terms of polynomials about an arbitrary point. In 1715 he published this method in Methodus in crementorum directa et inversa and so this method was named in his honor, Taylor's series. The Taylor's series is just a generalized form of the Maclaurin series formulated by the Scottish mathematician Colin Maclaurin in about 1742. Although Taylor published his method for expanding functions in 1715, James Gregory (sometimes spelled Gregorie) and J. Bernoulli knew these methods long before. Joseph-Louis Lagrange isolated Taylor series as the idea fundamental to calculus. He considered these expansions so important to calculus that he maintained that in order to understand a continuous function one needed to know only the derivatives of a function at a given number of points. His error was not having a clear concept of convergence and realizing that all continuous functions have a Taylor's series. Later it became recognized that the Taylor series did not provide the sole key to understanding continuous functions. In spite of limitations, Taylor's series have had a major impact on the development of calculus.