When Dan Kalman (mathematics and statistics) receives the Mathematical Association of America's (MAA) Lester R. Ford Award this summer, he won't be a stranger. Kalman's ninth prize for expository books and articles is a recognition of his special brand of scholarship, combining original mathematical thought, historical and literature research, and an investigative mind-set.
His latest award-winning paper, "Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function," published by the American Mathematical Monthly in 2012, was coauthored with Mark McKinzie.
The paper joins a list of Kalman's sometimes quirkily titled MAA winners, among them "Harvey Plotter and the Circle of Irrationality," written with Nathan Carter, and Kalman's 2008 book, Uncommon Mathematical Excursions: Polynomia and Related Realms, for which he also won the Beckenbach Book Prize.
His latest MAA award bookends his first. His 1993 prize-winning paper, playing off the title of a Paul Simon song, was called "Six Ways to Sum a Series."
Kalman's most recent article on summing the same series explores a variation on the theme, relating it to the work of legendary mathematician Leonhard Euler (1707–1783). Summing a series, Kalman explains, turns on a basic question: Can you add up an infinite number of things?
"It will come as a surprise to a number of people in the lay world that it is quite reasonable and plausible to add an infinite number of things," he says. "Here's an example. Suppose you have a cake and there's just one piece left. But you don't want to be greedy, so you cut that piece in half and just take that. The next person comes and cuts that in half and just takes that half. The next person comes and just eats half."
Each step could have a number. After a whole piece there would be half, then half of a half, then half of a quarter, and on and on.
"So here's where the leap of faith comes," Kalman continues. "Imagine now doing this an infinite number of times. So what would that mean? When you added up all those pieces what they have to add up to is the one piece you started with. So you get this type of equation that says if you take up this infinite number of fractions and add them up, the answer has to be one. It certainly can't be bigger than one; you can't have more cake than you had to begin with, and it can't be less than one because if there was any amount left over you could still cut it in half and take another piece out. So mathematicians long ago decided you can add an infinite number of things and have it equal to one." (In symbols, .)
In 1735, Kalman and McKinzie write in their paper, Euler arrived at a summation formula that startled his contemporaries: . This gives an exact value for an infinite summation that had defied mathematicians for decades, and the appearance of pi took everyone by surprise.
Kalman and McKenzie take as their task another method of arriving at through elementary calculus and finding ways around "roadblocks" one encounters using their method. They then explore a historical puzzle: Did Euler know this method, or given his previous work did he simply not think it worth mentioning?
Beyond the intellectual challenge of such exercises, Kalman says they also have real-world applications.
"Mathematics pursued because it's interesting almost inevitably ends up being important in application," he says. "In today's world, infinite series are used ubiquitously. We take complicated phenomena in nature and we approximate them using infinite summations. And there's a way to do this in a systematic way, and these things that we're adding up to approximate them behave very nicely and theoretically in mathematical terms. So you can analyze them, you can find things like error terms, and you can find approximations. When you digitize a voice message or a piece of music or a video, all of those things are changed into a digital format and they're described by numbers. And the methods people use to analyze those numbers—to know how can I be sure that I'll reproduce this with a small enough error that your ear can't hear the difference between this and the original"—are based on summing infinite series.
The Mathematical Association of America will present the Paul R. Halmos-Lester R. Ford Award to Kalman on August 2 at the MathFest 2013 Prize Session in Hartford, Connecticut. The awards are for the best articles of expository mathematics published in the American Mathematical Monthly.