The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
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Maclaurin's theorem is a specific form of Taylor's theorem, or a Taylor's power series expansion, where c = 0 and is a series expansion of a function about zero. The basic form of Taylor's theorem is: n = 0 (f(n)(c)/n!)(x - c)n. When the appropriate substitutions are made Maclaurin's theorem is: f (x) = f(0) + f'(0)x + f''(0)x2/2! + f(3)(0)x3/3! + ... f(n)(0)xn/n! +.... The Taylor's theorem provides a way of determining those values of x for which the Taylor series of a function f converges to f(x).
In 1742 Scottish mathematician Colin Maclaurin attempted to put calculus on a rigorous geometric basis as well as give many applications of calculus in the work. It was the first logical and systematic exposition of the method of fluxions and originated as a reply to Berkeley's attack on Newton's methods of calculus. In this text, among several other monumental ideas, Maclaurin gave a proof of the theorem that today holds his name, Maclaurin's theorem, and is a special case of Taylor's theorem. He obtained this theorem by assuming that f(x) can be expanded in a power series form and then, upon differentiation and substituting x = 0 in the results, the values of the coefficients of each term can be obtained. He did this but did not investigate the convergency of the series at that time. Although this theorem holds Maclaurin's name it was previously published by another Scottish mathematician, James Stirling, in his book Methodus Differentialis published in 1730. It is no doubt but that this theorem is what Maclaurin is best remembered.