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2.5 Taylor SeriesInfinite series explode across the eighteenth century. They are discovered, investigated, and utilized. They are recognized as a central pillar of calculus, so much so that one of the most important books to be published in this century, Euler's Introductio in analysin infinitorum of 1748, is a primer on infinite series. There is no calculus in it in the sense that there are no derivatives, no integrals, only what Euler calls “algebra,” but it is the algebra of the infinite: derivations of the power series for all of the common functions and some extraordinary manipulations of them. This is done not as a consequence of calculus but as a preparation for it. As he says in the Preface:
By the end of the seventeenth century, power series,
had emerged as one of the primary tools of calculus. They were useful for finding approximations. They soon became indispensable for solving differential equations. As long as x is restricted to the interval where the power series are defined, they can be differentiated, integrated, added, multiplied, and composed as if they were ordinary polynomials. One example of their utility can be found in Leonhard Euler's analysis of 1759 of the vibrations of a circular drumhead. Euler was led to the differential equation
where u (the vertical displacement) is a function of r (the distance from the center of the drum) and and are constants depending on the properties of the drumhead. There is no closed form for the solution of this differential equation, but if we assume that the solution can be expressed as a power series,
Power series are useful. They are also ubiquitous. Every time a power series representation was sought, it was found. It might be valid for all x as with sin x, or only for a restricted range of x as with ln(1+x), but it was always there. In 1671, James Gregory wrote to John Collins and listed the first five or six terms of the power series for
Clearly, he was drawing on some underlying machinery to generate these. Everyone seemed to know about this power series machine. Gottfried Leibniz and Abraham de Moivre had each described it and explained the path of their discovery in separate letters to Jean Bernoulli, Leibniz in 1694, de Moivre in 1708. Newton had hinted at it in his geometric interpretation of the coefficients of a power series in Book II, Proposition X of the Principia of 1687. He elucidated it fully in an early draft of De Quadratura but removed it before publication. Jean Bernoulli published the general result
in the journal Acta Eruditorum in 1694. He would
later point out that this is equivalent to the machine in question. Today,
this machine is named for the first person to actually put it into print,
Brook
Taylor (1685–1731). It appeared in his Methodus incrementorum
of 1715. His derivation is based on an interpolation formula discovered
independently by James Gregory and Isaac Newton (Book III, Lemma V of
the Principia). 
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