Abstract
Apparently it started with a discussion in Child’s Coffeehouse where Brook Taylor (1685–1731) got the idea for the now famous series. He was talking with his friend John Machin about solving Kepler’s problem. As it turned out, the Taylor series was of such importance that Lagrange called it “the basic principle of differential calculus.” Indeed, it plays a very important part in calculus as well as in computation, statistics, and econometrics. As it is well known, a calculator or computer can only add and, in fact, can deal only with 0 s and 1 s. So how is it possible that you punch in a number and then press a button, and the calculator finds the logarithm or exponential of that number? Similarly, how can a machine capable of only adding give you the sine and cosine of an angle, find solutions to an equation, and find the maxima and minima of a function? All these and more can be done due to the Taylor series.
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Notes
- 1.
The Taylor expansion around point 0 is referred to as Maclaurin expansion after Colin Maclaurin (1698–1746), a brilliant mathematician who derived it as a special case of Taylor series.
- 2.
The self-taught French mathematician Michel Rolle (1652–1719) is best known for this theorem.
- 3.
After Joseph-Louis Lagrange (1736–1813), who was considered a great mathematician at 23 and whom Napoleon Bonaparte referred to as “The Lofty Pyramid of the mathematical sciences.”
- 4.
Sir Isaac Newton (1643–1727) is a giant in the history of science; indeed, the publication of his Philosophiae Naturalis Principia Mathematica, usually referred to as Principia, is a turning point in the history of humankind. He invented calculus, discovered important laws of physics, and showed that the universe works on mathematical principles. Yet, he found time to improve the operation of the Royal Mint. He also served as the president of the Royal Society and shaped it to become the leading scientific society in the world. Newton was a loner and secretive. He did not acknowledge the contribution of other scientists and got into a bitter dispute with Leibnitz over the invention of calculus and with Robert Hooke (1635–1703), another pioneer scientist, over the theory of light. There are many good books on the history of science. I suggest John Gribbin’s Science, A History 1543–2001 (2002). On the life of Newton, the reader may be interested in reading Isaac Newton: The Last Sorcerer by Michael White (1997). The German mathematician, Gottfried Wilhelm Leibnitz (1646–1716), independently invented calculus. His exposition was easier to understand than Newton’s. The two engaged in a bitter dispute over who had priority. Leibnitz organized the Berlin Academy of Sciences and served as its first president. He had other interests, including law and economics, and for a time served as a diplomat.
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Dadkhah, K. (2011). The Taylor Series and Its Applications. In: Foundations of Mathematical and Computational Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13748-8_10
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