Download PDF by Daniel J. Velleman: American Mathematical Monthly, volume 117, number 4, April

By Daniel J. Velleman

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Extra resources for American Mathematical Monthly, volume 117, number 4, April 2010

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378], but I don’t know any other sources for it. 1) + sin(c − f ) sin(c − g) sin(c − h) sin( f − a) sin( f − b) sin( f − c) + sin(c − a) sin(c − b) sin( f − g) sin( f − h) + sin(g − a) sin(g − b) sin(g − c) sin(h − a) sin(h − b) sin(h − c) + = 0. sin(g − f ) sin(g − h) sin(h − f ) sin(h − g) This follows (he said, still quoting Glaisher) since the first three terms add up to sin(a + b + c − f − g − h), and the last three to sin( f + g + h − a − b − c). He noted that Glaisher made these assertions sans d´emonstration at a conference in Reims in 1880.

There are a number of interesting exercises involving quasi-Cauchy sequences that can be given to undergraduate analysis or topology students: 1. Suppose that f is uniformly continuous on some interval I . Prove that the image under f of any quasi-Cauchy sequence in I is quasi-Cauchy. 2. Suppose that xn and yn are quasi-Cauchy sequences of real numbers. Prove or give a counterexample to the claim that the product sequence xn yn is also quasi-Cauchy. 3. Prove that a sequence xn of real numbers is Cauchy if and only if every subsequence is quasi-Cauchy.

J. L¨utzen, Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Springer-Verlag, New York, 1990. T. Muir, The Theory of Determinants in the Historical Order of Development, vol. 3, Macmillan, London, 1920; reprinted by Dover, New York, 1960. G. Polya, Mathematics and Plausible Reasoning, vol. 1, Induction and Analogy in Mathematics, Princeton University Press, Princeton, NJ, 1954. R. Remmert, Theory of Complex Functions (trans. R. Burckel), Graduate Texts in Mathematics, vol. 122, Springer-Verlag, New York, 1991.

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