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Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function

Year of Award: 2013

Award: Halmos-Ford Award

Publication Information: The American Mathematical Monthly, Vol. 119, 2012, pp. 42-51

Summary: (Adapted from the MathFest 2013 Prizes and Awards Booklet)

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About the Author: (From the MathFest 2013 Prizes and Awards Booklet)

Dan Kalman has been a member of the mathematics faculty at American University, Washington, DC since 1993. Prior to that he worked for eight years in the aerospace industry and taught at the University of Wisconsin, Green Bay. Kalman has a B.S. from Harvey Mudd College and a Ph.D. from University of Wisconsin, Madison. He has been a frequent contributor to all of the MAA journals, has published two books with the MAA, and has served the MAA in various capacities, including as an MAA associate executive director, on the MAA Board of Governors, on the editorial boards of both MAA book series and journals, and as a cast member of both productions of MAA - the Musical.

Mark McKinzie earned his Ph. D. in mathematics from the University of Wisconsin in 2000. His dissertation on formalist techniques in the early history of power series was completed under the supervision of Michael Bleicher and fostered an ongoing fascination with the mathematical work of Edmond Halley and Leonhard Euler. He taught at Monroe Community College for five years, and in 2004 joined the Department of Mathematical and Computing Sciences at St. John Fisher College. When not thinking about mathematics and its history, he enjoys traveling with his family, learning new recipes and cooking techniques, and solving kakuru puzzles.

MSC Codes: 
01A50, 40G05
Dan Kalman and Mark McKinzie
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Publication Date: 
Monday, August 19, 2013
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The authors investigate a proof, different from the usual ones, for the result that the sum of the reciprocal squares is \(\pi^2/6\). This approach to the sum might have been known to Euler. After sketching the details as might have been done by Euler, the authors give a contemporary justification of the proof. Kalman and McKinzie end by raising the question of whether Euler actually knew of this argument.