In mathematics the Taylor series of a function is an infinite sum of terms that are expressed in terms of the functions derivatives at a single point For most common functions the function and the sum of its Taylor series are equal near this point Taylors 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 n first 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 Taylors 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 The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result but rejected it as an impossibility2 the result was Zenos paradox Later Aristotle proposed a philosophical resolution of the paradox but the mathematical content was apparently unresolved until taken up by Archimedes as it had been prior to Aristotle by the Presocratic Atomist Democritus It was through Archimedess method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result3 Liu Hui independently employed a similar method a few centuries later4 In the 14th century the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama56 Though no record of his work survives writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series including those for the trigonometric functions of sine cosine tangent and arctangent The Kerala School of Astronomy and Mathematics further expanded his works with various series expansions and rational approximations until the 16th century In the 17th century James Gregory also worked in this area and published several Maclaurin series It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor7 after whom the series are now named The Maclaurin series was named after Colin Maclaurin a professor in Edinburgh who published the special case of the Taylor result in the 18th century