Beyond Mere Convergence Jaes A. Sellers Departent of Matheatics The Pennsylvania State University 07 Whitore Laboratory University Park, PA 680 sellers@ath.psu.edu February 5, 00 REVISED Abstract In this article, I suggest that calculus instruction should include a wider variety of exaples of convergent and divergent series than is usually deonstrated. In particular, a nuber of convergent series, such as k 3, are considered, and their exact values are found in a k straightforward anner. We explore and utilize a nuber of atheatical topics, including anipulation of certain power series and recurrences. During y ost recent spring break, I read Willia Dunha s book Euler: The Master of Us All 3]. I was thoroughly intrigued by the aterial presented and a certainly glad I selected it as part of the week s reading. Of special interest were Dunha s coents on series anipulations and the power series identities developed by Euler and his conteporaries, for I had ust copleted teaching convergence and divergence of infinite series in y calculus class. In particular, Dunha 3, p. 47-48] presents Euler s proof of the Basel Proble, a challenge fro Jakob Bernoulli to deterine the
exact value of the su. Euler was the first to solve this proble by k proving that the su equals π. 6 I was reinded of y students interest in this result when I shared it with the ust weeks before. I had already entioned to the that exact values for relatively few failies of convergent series could be deterined. The obvious exaples are geoetric series r k (with r < ) and telescop- k 0 ing series. I also reebered their disappointent when I observed that the exact nuerical value of ost convergent series cannot be deterined in a straightforward way. I tried to excite the with the notion that the convergence or divergence of a given series could be deterined via the Integral Test, Liit Coparison Test, Ratio or Root Test, but this was received with little enthusias. But now I return to Dunha s book. In 3, p. 4], Dunha notes that Jakob Bernoulli, p. 48-49] proved () and () k k = 6 k 3 k = 6. Many teachers of calculus will recognize at least two things about () and (). First, these series are ade-to-order exaples to deonstrate convergence with the Ratio Test. Such exaples, where the suands are defined by the ratio of a polynoial and an exponential function, can be found in a nuber of calculus texts, such as 4] and 5]. Second - a uch ore negative adission - is that we rarely teach students how to prove equalities like () and (). We usually stop at deonstrating that such series converge, and ove on to other atters. This is the case with the two calculus texts entioned above, and it is an unfortunate situation to say the least. I contend that students of first-year calculus would be better served if we provided a few ore tools to the for finding exact values of convergent infinite series. Oddly enough, the series in () and () are ideal for such a task.
My goal in this note is to present two approaches to finding the exact value of a(, n) := k n k with > and n N {0} (of which Bernoulli s exaples () and () are special cases). We begin by noting that, for each >, <, so that a(, 0) is a convergent geoetric series. Moreover, k a(, 0) = = + ( k = + ) k ( = + a(, 0). ) k Solving for a(, 0), we see that it equals. Of course, this result easily follows fro the usual forula for the su of a convergent geoetric series. Next, we obtain a recurrence for a(, n), n, in ters of a(, ) for < n. Note that a(, n) = k n k = + k n k k = + (k + ) n k = + ] (k + ) n. k 3
The arguent up to this point is exactly that used in finding the forula for a(, 0) above. We now eploy the binoial theore, a tool that should be in the repertoire of first-year calculus students. ( n ) k ) a(, n) = + = n + = n + = n + ( n k ( ) n ) ( n = a(, n) + ] k k k k + k n k ( ) n a(, ) + a(, n) n + ] ] ( ) ] n a(, ) Solving for a(, n) yields ( ) a(, n) = n ( ) ] n + a(, ) or (3) a(, n) = ( ) n ( ) ] n + a(, ). As a sidenote, it is interesting to see fro (3) that, for rational values of, the nuerical value of a(, n) ust be rational for all n 0. This can be proven via induction on n. We noted above that a(, 0) = which is rational as long as is rational. Then, assuing a(, ) is rational for 0 n, (3) iplies a(, n) is also rational. Hence, no values such as π will arise as values for a(, n) whenever is rational. 6 The recurrence in (3) can be used to calculate with relative ease the exact value of a(, n) = 4 k n k
for all > and n N {0}. For exaple, since a(, 0) = k =, we have and a(, ) = k k ( = = + =, k ) + a(, ) = k ( ) = + a(, 0) + 0 = + + = 6, ( ) ] a(, 0) 0 ( ) a(, ) which is the result labeled (). Finally, k 3 a(, 3) = k ( ) ( ) 3 3 = + a(, 0) + a(, ) + 0 = + + 3 + 3 6 = 6, ( ) 3 a(, ) which is (). Of course, recurrence (3) could be used to calculate a(, n) for larger values of and n. However, this ight prove tedious for extreely large values of n. With this in ind, we now approach the calculation of a(, n) fro a second point of view. We begin with the failiar power series representation for the function x : (4) x = + x + x + x 3 + x 4 +..., where x < 5
Andrews ] recently extolled the virtues of (4) in the study of calculus. Our goal in this section is to anipulate (4) via differentiation and ultiplication to obtain a new power series of the for f n (x) := x + n x + 3 n x 3 + 4 n x 4 +... = k n x k for a fixed positive integer n. This is done ( by ) applying the x d operator to n ties. Then a(, n) equals f x n, which is easily coputed once f n (x) is written as a rational function. (Note that we define f 0 (x) by f 0 (x) := x ( ) x = x k.) or Hence, As an exaple, we apply the x d operator to x and get x d ( ) = x d x ( + x + x + x 3 + x 4 +...) f (x) = We can apply the x d Thus, x ( x) = x + x + 3x 3 + 4x 4 +... = kx k. k k = f ( ) = ( ) =. operator to twice to obtain f x (x) : f (x) = x d ( x d ( )) x ( ) = x d = x + x ( x) 3. x ( x) f (x) = x + x ( x) = x + 3 x + 3 x 3 + 4 x 4 +... = k x k. 6
Hence, k k = f ( ) = ) + ( ( ) 3 = 6 upon siplification. This, as we have already seen, is (). Additional applications of the x d operator can be perfored to yield f (x) = x ( x) = kx k, f (x) = x + x ( x) = k x k, 3 f 3 (x) = x3 + 4x + x = k 3 x k, ( x) 4 f 4 (x) = x4 + x 3 + x + x = k 4 x k, ( x) 5 f 5 (x) = x5 + 6x 4 + 66x 3 + 6x + x = k 5 x k, and ( x) 6 f 6 (x) = x6 + 57x 5 + 30x 4 + 30x 3 + 57x + x = k 6 x k. ( x) 7 We see that f n (x) = g n(x) ( x) n+ for each n where g n (x) is a certain polynoial of degree n. Indeed, the functions g n (x) are well-known. Upon searching N.J.A. Sloane s On-Line Encyclopedia of Integer Sequences 6] for the sequence,,,, 4,,,,,,, 6, 66, 6,,..., which is the sequence of coefficients of the polynoials g n (x), we discover that these are the Eulerian nubers e(n, ). They are defined, for each value of and n satisfying n, by (5) e(n, ) = e(n, ) + (n + )e(n, ) with e(, ) =. 7
With this notation, it appears that, for n, f n (x) = n e(n, )x = ( x) n+. Using (5), this assertion can be proven in a straightforward anner via induction. Moreover, we know fro 6, Sequence A0089] that e(n, ) = ( ) n + ( ) l ( l) n. l l=0 This can be used to write the rational version of f n (x) for any n in a tiely way. So, for exaple, we see that f 8 (x) = x8 + 47x 7 + 493x 6 + 569x 5 + 569x 4 + 493x 3 + 47x + x ( x) 9, which iplies k 8 5 k = f 8 ( ) = 39685 5 048. We have thus seen two different ways to copute the exact value of k n with > and n N {0}, one with a recurrence and one k with power series. I encourage us all to share at least one of these techniques with our students the next tie we are exploring infinite series. References ] G. Andrews, The Geoetric Series in Calculus, Aerican Matheatical Monthly 05, no. (998), 36-40. ] J. Bernoulli, Tractatus de seriebus infinitis, 689. 3] W. Dunha, Euler: The Master of Us All, The Dolciani Matheatical Expositions, no., Matheatical Association of Aerica, Washington, D.C., 999. 8
4] C. Edwards and D. Penney, Calculus with Analytic Geoetry, Fifth Edition, Prentice Hall, 998. 5] R. Larson, R. Hostetler, and B. Edwards, Calculus: Early Transcendental Functions, Second Edition, Houghton Mifflin Copany, 999. 6] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.co/ nas/sequences/. Keywords: infinite series, convergence, divergence, Euler, Bernoulli, ratio test, recurrence, binoial theore, Eulerian nubers Biographical Note: Jaes A. Sellers is currently the Director of Undergraduate Matheatics at the Pennsylvania State University. Before accepting this position he served for nine years as a atheatics professor at Cedarville University in Ohio. As a atheatics professor, Jaes loves to teach atheatics to undergraduates and perfor research with the. Prior to going to Cedarville he received his Ph.D. in atheatics in 99 fro Penn State, where he et his wife Mary. Jaes truly enoys spending tie with Mary and their five children. He agrees with Euler that atheatics can often be enoyed and discovered with a child in his ars or playing round his feet. 9