Mark McKinzie

Euler and a modern evaluation of \(1+\frac{1}{4}+\frac{1}{9}+\cdots \)

Abstract:

The story of Euler's original evaluation of \(\zeta(2)=1+\frac{1}{4}+\frac{1}{9}+\cdots \) and subsequent rederivations is well known. Each derivation shows the familiar Euler genius for creative manipulation of series. In the modern classroom, it is tempting to attempt an evaluation of the series by more mundane means, by manipulating the power series \(\sum z^k/k^2\). One progresses without difficulty to find that \(\zeta(2)=\int_0^1 -\frac{\ln(1-t)}{t}dt\). Evaluating this integral presents an obstacle, but success is possible if one is aware of some properties of the dilogarithm function \(\text{Li}_2(z)=\int_0^z -\frac{\ln(1-t)}{t}dt\). Following this path, we will obtain another derivation of the value of the series. So far as I know, the earliest appearance in print of this particular method for evaluating the series is from c. 1980.

Curiously, the necessary properties of the dilogarithm were first discovered by Euler himself. His initial work on the dilog function predates his evaluation of \(\sum 1/n^2 \), appearing in a 1730 paper where he estimates the value of the series to six decimal places. The critical identity for this new derivation was published by Euler in a 1779 paper. In that paper, Euler might easily have then evaluated \(\zeta(2)\), but instead he takes its value to be a well-known result. Did Euler realize that his methods provided yet another way to compute \(\sum 1/n^2 \)? Could he have failed to notice?

(This talk is based on joint work with Dan Kalman at American University.)

Biography:

Mark McKinzie earned his Ph.D. in mathematics from the University of Wisconsin in 2000. His dissertation, on the early history of power series, kindled a fascination with the mathematical work of Edmond Halley and Leonhard Euler, and the history of mathematics more generally. He was an Instructor in the Mathematics Department at Monroe Community College from 1999 to 2004, and is currently an Associate Professor at St. John Fisher College in the Department of Mathematical and Computing Sciences. Mark co-authored two papers which were recognized by the MAA with writing awards, the Carl B. Allendoerfer Award (2002), and the Paul R. Halmos - Lester R. Ford Award (2013).