Projects & Research A Radical Approach to Real Analysis Macalester College


Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

>2.5 Taylor Series

Taylor's Formula


Lagrange's Remainder

2.6 Emerging Doubts

2.5 Taylor Series

Infinite 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:

“Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use… I am certain that the material I have gathered in this book is quite sufficient to remedy that defect.”

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,

then we can solve for and the .

Click here to see Euler's solution of the differential equation.

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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