NOTES ON SECTION 1 ------------------ The infinite product given for (pi/2) is attributed to Wallis. The coninued fraction given for (e-1) is attributed to Euler. The continued fraction given for (4/pi) is attributed to Brouncker. A nested radical is given with limit K. K is sometimes called the Kasner number, and is approximately 1.757933. See [4] for further information. NOTES ON SECTION 2.3 -------------------- The series given for (2/pi) was originally discovered by Bauer before being rediscovered by Ramanujan (see page 167 of [7]). NOTES ON SECTION 4 ------------------ [10] contains an introduction to analysis of infinite products. The two identities given in subsections A) and B) can be found in [9]. The statement made in subsection C) can be found in [1], which gives an analogous statement for infinite exponential ladders. For information on Exponential Ladders, see [1], [2], [3], [11]. For information on Herschfeld's Convergence Theorem, see [3], [4], [6]. For information on the "souped-up" ratio test for continued powers, see [5]. NOTE ON LAST SLIDE ------------------ This picture was taken from page 13 of "Mathematical Impressions", by A.T. Fomenko, 1991. (Davis Center call number NC137.F66A4) REFERENCES ---------- [1] Barrow, D.F., "Infinite Exponentials", American Mathematical Monthly, volume 43, pages 150-160, March 1936. (Davis Center call number QA1.A5) [2] Creutz, Michael and Sternheimer, R.M., "On the Convergence of Iterated Exponentiation", Fibonacci Quaterly, number 18, December 1980. (Davis Center call number QA241.F5) [3] Eric's treasure trove of math: http://www.astro.virginia.edu/~eww6n/math/ [4] Herschfeld, A., "On Infinite Radicals.", The American Mathematical Monthly, volume 42, pages 419-429, 1935. (Davis Center call number QA1.A5) [5] Jones, Dixon J., "Continued Powers and Roots", Fibonacci Quaterly, number 29, February 1991. (Davis Center call number QA241.F5) [6] Jones, Dixon J., "Continued Powers and a Sufficient Condition for Their Convergence", Mathematics Magazine, volume 68, number 5, pages 387-392, December 1995. (Davis Center call number QA1.N35) [7] Kanigel, Robert, "The Man Who Knew Infinity (A Life of the Genius Ramanujan)", Washington Square Press, 1991. [8] Knopfmacher, Arnold and Knopfmacher, John, "A New Infinite Product Representation for Real Numbers", Monatshefte fur Mathematik, number 104, pages 29-44, 1987. (Davis Center call number QA1.M877) [9] Knopfmacher, Arnold and Knopfmacher, John, "A New Construction of the Real Numbers (via infinite products)", Nieuw Archief Voor Wiskunde, number 5, pages 19-31, 1987. (Davis Center call number QA1.N5) [10] Marsden, Jerrold E., "Basic Complex Analysis", WH Freeman & Co., 2nd edition, 1987. (Davis Center call number QA331.M378) [11] Simon, Louis, "Sur les exponentielles superposées, propriétés générales, prolongements fonctionnels.", Paris, A. Blanchard, 1966. (Davis Center call number QA161.S5)