Editing Taylor series (section)

== History ==
The [[ancient Greek philosopher]] [[Zeno of Elea]] considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;{{sfn|Lindberg|2007|p=33}} the result was [[Zeno's 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 Archimedes's [[method of exhaustion]] that an infinite number of progressive subdivisions could be performed to achieve a finite result.{{sfn|Kline|1990|p=[https://archive.org/details/mathematicalthou00klin/page/n437 35]–37}} [[Liu Hui]] independently employed a similar method a few centuries later.{{sfn|Boyer|Merzbach|1991|p=[https://archive.org/details/historyofmathema00boye/page/202 202–203]}}<!--I'm sure there are better refs than this. Hui gave fairly "rigorous" bounds on the convergence, if I recall. But it isn't addressed here.-->

In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician [[Madhava of Sangamagrama]].{{sfn|Dani|2012}} Though no record of his work survives, writings of his followers in the [[Kerala school of astronomy and mathematics]] suggest that he found the Taylor series for the [[trigonometric function]]s of [[sine]], [[cosine]], and [[arctangent]] (see [[Madhava series]]). During the following two centuries his followers developed further series expansions and rational approximations.

In late 1670, [[James Gregory (mathematician)|James Gregory]] was shown in a letter from [[John Collins (mathematician)|John Collins]] several Maclaurin series {{nobr|(<math display=inline>\sin x,</math>}} <math display=inline>\cos x,</math> <math display=inline>\arcsin x,</math> and {{nobr|<math display=inline>x \cot x</math>)}} derived by [[Isaac Newton]], and told that Newton had developed a general method for expanding functions in series. Newton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself. In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for <math display=inline>\arctan x,</math> <math display=inline>\tan x,</math> <math display=inline>\sec x,</math> <math display=inline>\ln\, \sec x</math> (the integral of {{nobr|<math>\tan</math>),}} <math display=inline>\ln\, \tan\tfrac12{\bigl(\tfrac12\pi + x\bigr)}</math> (the [[integral of the secant function|integral of {{math|sec}}]], the inverse [[Gudermannian function]]), <math display=inline>\arcsec \bigl(\sqrt2 e^x\bigr),</math> and <math display=inline>2 \arctan e^x - \tfrac12\pi</math> (the Gudermannian function). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671.<ref>{{multiref|{{harvnb|Turnbull|1939|pp=168–174}}|{{harvnb|Roy|1990}}|{{harvnb|Malet|1993}}}}</ref>

In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work ''De Quadratura Curvarum''. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title ''Tractatus de Quadratura Curvarum''.

It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by [[Brook Taylor]],<ref>{{multiref|{{harvnb|Taylor|1715|p=21–23, see Prop. VII, Thm. 3, Cor. 2}}. See {{harvnb|Struik|1969|pp=329–332}} for English translation, and {{harvnb|Bruce|2007}} for re-translation.|{{harvnb|Feigenbaum|1985}}}}</ref> after whom the series are now named.

The Maclaurin series was named after [[Colin Maclaurin]], a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.