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Taylor's Theorem

Taylor's theorem states that any function satisfying certain conditions may be represented by a Taylor series,

Taylor's theorem (without the remainder term) was devised by Taylor in 1712 and published in 1715, although Gregory had actually obtained this result nearly 40 years earlier. In fact, Gregory wrote to John Collins, secretary of the Royal Society, on February 15, 1671, to tell him of the result. The actual notes in which Gregory seems to have discovered the theorem exist on the back of a letter Gregory had received on 30 January, 1671, from an Edinburgh bookseller, which is preserved in the library of the University of St. Andrews (P. Clive, pers. comm., Sep. 8, 2005).

However, it was not until almost a century after Taylor's publication that Lagrange and Cauchy derived approximations of the remainder term after a finite number of terms (Moritz 1937). These forms are now called the Lagrange remainder and Cauchy remainder.

Most modern proofs are based on Cox (1851), which is more elementary than that of Cauchy and Lagrange (Moritz 1937), and which Pringsheim (1900) referred to as "leaving hardly anything to wish for in terms of simplicity and strength" (Moritz 1937).

Cauchy Remainder, Lagrange Remainder, Taylor Series

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References

Cox, H. Cambridge and Dublin Math. J. 6, 80, 1851.Dehn, M. and Hellinger, D. "Certain Mathematical Achievements of James Gregory." Amer. Math. Monthly 50, 149-163, 1943.Jeffreys, H. and Jeffreys, B. S. "Taylor's Theorem." §1.133 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 50-51, 1988.Malet, A. Studies on James Gregorie (1638-1675). Ph.D. thesis. Princeton, NJ: Princeton University, 1989.Malet, A. "James Gregorie on Tangents and the 'Taylor' Rule for Series Expansions." Archive for History of Exact Science 46, 97-137, 1993-1994.Moritz, R. E. "A Note on Taylor's Theorem." Amer. Math. Monthly 44, 31-33, 1937.Pringsheim, A. "Zur Geschichte des Taylorschen Lehrsatzes." Bibliotheca Math. 1, 433-479, 1900.Todhunter, I. A Treatise on the Differential Calculus with Numerous Examples, 10th ed. London: Macmillan, p. 75, 1890.Turnbull, H. W. (Ed.). James Gregory: Tercentenary Memorial Volume. London: Bell, 1939.

Taylor's Theorem

Cite this as:

Weisstein, Eric W. "Taylor's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TaylorsTheorem.html