Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function Dan Kalman and Mark McKinzie Abstract. It is tempting to try to reprove Euler’s famous result that P 1=k2 D π 2=6 using R 1 − ln.1−t/ power series methods of the sort taught in calculus 2. This leads to 0 t dt, the evalua- tion of which presents an obstacle. With two key identities the obstacle is overcome, proving the desired result. And who discovered the requisite identities? Euler! Whether he knew of this proof remains to be discovered. It is by now a familiar story: the young Leonhard Euler stuns the mathematical world in 1735 by announcing that P 1=k2 D π 2=6. (Here and throughout, the indices in summations are assumed to run from 1 to 1.) As Dunham [2] explains, the problem had been open since 1644, “and anyone capable of summing the series was certain to make a major splash.” Sandifer [13] gives a detailed account of Euler’s several derivations of the π 2=6 result, but our favorite retelling is the succinct statement of Erdos˝ and Dudley [3], “In 1731 he obtained the sum accurate to 6 decimal places, in 1733 to 20, and in 1734 to infinitely many.” Today many different proofs of Euler’s result are known. Weisstein [15] lists nearly a dozen references to proofs, and Kalman’s survey [9] gives six proofs in detail. So, between the historical accounts of Euler’s solution and all of the known proofs, can there be anything more to say about the subject? We say there is. In this paper we will present yet another method for finding P 1=k2, which deserves consideration for several reasons. First, the analysis starts with methods of elementary calculus, of the sort that any second-semester calculus student might think to try. Second, after what appears to be encouraging preliminary success, the method runs into a roadblock. It turns out that there is a way ’round, relying on two key identities. And whom may we thank for those identities? None other than Euler. Indeed, the proof possesses that familiar Eulerian flourish, with deli- cious manipulations and a breezy disregard for technicalities. But don’t worry. All the manipulations can be justified, leaving us with a rigorous determination of P 1=k2. We are also left with a historical puzzle. Did Euler know this proof? As we will show, though the historical evidence is inconclusive, he might well have known it. Given Euler’s genius, it is hard to imagine that he did not know it. He certainly had all the necessary steps well in hand. But in the paper containing the key identity, there is a hint that Euler either was not aware of this proof or did not consider the proof worth mentioning. Here we will take up each of these matters: a second-semester calculus approach to evaluating P 1=k2, the roadblock that arises, the Euler-style manipulations that get us past the roadblock, and the rigorous justification for the manipulations, as well as a brief look at our historical puzzle. Before we proceed a few comments are in order. First, we acknowledge Lewin [12, p. 4] as our source for the derivation to follow. In fact, we shall refer to it as Lewin’s argument, although we are not certain that it originated with him. Second, in citing http://dx.doi.org/10.4169/amer.math.monthly.119.01.042 MSC: Primary 11M06, Secondary 33B30 42 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 119 Euler’s publications, we provide references to the most readily available editions of his works—i.e., the scanned images available online at the Euler Archive [4], as well as the Opera Omnia, the modern reprinting of his collected work. For each publication, we also provide the Enestrom¨ index number, a valuable aid regardless of which source one uses. In those cases where it is necessary to cite a specific passage in one of Euler’s works, paragraph or section numbers are used, as these are fairly consistent across different printings and translations. In retracing the history of Euler’s work, it is important to distinguish between a date of publication (which is included in the bibliographical citations) and the times when his discoveries were made. In some cases the publications themselves indicate a date of presentation before a learned society, frequently far in advance of the publication date. The Enestrom¨ index also specifies dates of completion for some works, and Eu- ler’s correspondence provides another means for dating his discoveries. In considering whether Euler was aware of Lewin’s argument, we will see that timing may be signif- icant. Where we specify dates of particular results, they are generally as reported in [1, 2, 13]. The Calculus 2 Approach and Roadblock. The alert calculus 2 student catches a glimpse of a powerful idea: generating functions. It is one of the most important tools in enumerative combinatorics and a bridge between discrete mathematics and con- tinuous analysis [16]. But even the tiny glimpse afforded our calculus 2 student is impressive enough. Turning a numerical series into a function, we apply the methods of calculus. This allows us to find not only the sum of the original series but the sums of an infinite number of related series. Here is an example: Suppose we wish to sum the alternating series X − 1 1 1 1 .−1/k 1 D 1 − C − C··· : k 2 3 4 Consider the closely related power series X 1 z2 z3 z4 f .z/ D zk D z C C C C··· : (1) k 2 3 4 The general problem of evaluating f is, at first glance, much harder than our original problem, which only concerns − f .−1/. But if we differentiate (1), we obtain the geometric series 0 X − f .z/ D zk 1 D 1 C z C z2 C··· : This tells us that f 0.z/ D 1=.1 − z/, and hence Z f .z/ D 1=.1 − z/ dz D − ln.1 − z/ C C: Moreover, we know from the definition that f .0/ D 0. Consequently, C D 0 and so f .z/ D − ln.1 − z/. It follows that − f .−1/ D ln 2 and the original problem is solved. Not only that, we can evaluate f at any number of points to obtain sums of other series. For example, P 1=.k3k / D f .1=3/ D − ln.1=3/ D ln 3. There is a small fly in the ointment. The radius of convergence for f is 1, justifying term-by-term differentiation for −1 < z < 1. Our evaluation of P 1=.k3k / is perfectly valid. We have to work a little harder to justify applying our results when z D −1. But January 2012] ANOTHER WAY TO SUM A SERIES 43 let us not lose heart. For the original series 1 − 1=2 C 1=3 − 1=4 C··· ; numerical investigation must surely convince the most determined skeptic that ln 2 is correct. We can worry about the technical justification later. Exhilarated by the success of this method, we charge forward. What other series might it sum for us? Nothing could be more natural than P 1=k2. So, beginning as before, our goal is to sum the series X 1 1 1 1 D 1 C C C C··· : k2 4 9 16 The related power series X 1 z2 z3 g.z/ D zk D z C C C··· (2) k2 4 9 has derivative 2 0 X 1 − z z g .z/ D zk 1 D 1 C C C··· : k 2 3 But this is evidently f .z/=z, with f defined as above. Therefore, g0.z/ D − ln.1 − z/=z: (3) Now all that remains is to integrate − ln.1 − z/=z. Our calculus 2 student knows several methods that appear promising. Perhaps integration by parts. Or what about a substitution? Unfortunately, although the integrand doesn’t seem especially compli- cated, none of our efforts lead anywhere. We have hit a road block. How frustrating! We have come tantalizingly close to summing the series. In fact, because we know that g.0/ D 0, we can express the sum we seek as a definite integral: Z 1 Z 1 X 1 0 ln.1 − z/ D g.1/ D g .z/ dz D − dz: 2 k 0 0 z If only we could evaluate the integral. Now we are confronted by two possibilities. Perhaps ln.1 − z/=z is one of those functions that simply cannot be antidifferentiated in closed form. If so, our roadblock is probably impassable. On the other hand, it may be that we have just not found the proper trick. Here today’s students have resources not available to Euler. They can use mathemat- ics software. If the required integral can be evaluated by elementary means, a symbolic integrator will almost certainly be able to show us the answer. Let us try Quickmath [11], a free website powered by Mathematica. We ask it to integrate -log(1-x)/x for x going from 0 to 1. The answer comes back: π 2=6. That’s encouraging. How did Quickmath do that? To find out, we ask for the indefinite inte- gral of -log(1-x)/x, and quickmath responds Li2(x). Investigating further, we learn that Li2 is the dilogarithm function, a special case of the polylogarithm function [14]. Alas, we also discover that dilog is defined by the very series we dubbed g.z/.