Another Elementary Proof of Euler's Formula for (2n)

Another Elementary Proof of Euler's Formula for ζ(2n) Author(s): Tom M. Apostol Source: The American Mathematical Monthly, Vol. 80, No. 4 (Apr., 1973), pp. 425-431 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2319093 Accessed: 13/10/2010 13:41 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=maa. 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Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org 1973] CLASSROOM NOTES 425 = sin 0n () ikcotn ko, k=Ok we obtain the trigonometric identity sin nO = sin ((7)cotn'- - () cotn3 + (n)ctn-5o-+ Take n = 2m + 1 and write this in the form = 2m+1 I t (3) sin(2m + 1)0 = sin OPm(cot2O) with 0 < 0 < 2 where Pm is the polynomial of degree m given by Pm(x) = (2 + 1 )xm - (2 + 1)xm- 1 + (2 + 1) xm-2- + Since sin 0 # 0 for 0 < 0 < it/2, equation (3) shows that Pm(cot20) = 0 if and only if (2m + 1)0 = kir for some integer k. Therefore Pm(x) vanishes at the m distinct points Xk = cot2 rk/(2m + 1) for k = 1,2, ***,m. These are all the zeros of Pm(x) and their sum is cot2 irk _ (2m + 1 2m + 1 m(2m-1) which proves (2). NOTE. This paper was translated from a Greek manuscript and communicated to the MONTHLY on behalf of the author by Tom M. Apostol, California Institute of Technology. After this paper was written it was learned that the same proof was discovered independently and published in Norwegian by Finn Holme in Nordisk Matematisk Tidskrift, vol. 18 (1970), pp. 91-92. See also A. M. Yaglom and I. M. Yaglom, Challenging mathematical problems with elementary solutions, vol. II, Holden-Day, San Francisco, 1967, problem 145. ANOTHER ELEMENTARY PROOF OF EULER'S FORMULA FOR C(2n) TOM M. APOSTOL, California Institute of Technology 1. Introduction. The classic formula 10 - ) (27r)2nB2n (1) C(2n) = E = )n (2) which expresses C(2n) as a rational multiple of 7r2n was discovered by Euler [2]. The numbers Bn are Bernoulli numbers and can be defined by the recursion formula Bo= 1, Bn = ( )Bs for n 2, or equivalently, as the coefficients in the power series expansion 426 T. M. APOSTOL [April z B z n (2) e'-1 ! zj < 27r. In this notation we have (3) B1 =-I B2n+1 =O for n > 1, and 1 1 1 ~ ~ ~~~~~~1 5 B2= B4 = - B6= B8 - B1o = Euler's original proof of (1) was obtained from two distinct representations of iz cot JtZ, a power series expansion obtainable from (2), 7rz cotrzZ = 1 + E ( -)n( (2n)! 2n , valid for I z I < I and the partial fraction decomposition 0 2 itzcotrz = I-2 - 2 2' valid for z # 0, + 1 + 2,*. k1 k2 - Z If I z 1 < 1, each term in the last sum can be expanded in a geometric series giving us oo oo z 2 n oo irzcotzZ = I1-2 2 Y iL = 1-2 E C(2n)z2n. k=1 n= n = 1 Equation (1) follows by equating coefficients of 2n in the two power series expansions of irzcot irz. Details justifying this argument are given in Knopp [4], pp. 203-207, 236. Another well-known proof is obtained by putting s = 2n in Riemann's functional eq uation (l - s) = 2(27r)-S F(s)cos?;(s) and using the fact that C(1 - 2n) =-B2,,/(2n). These results are deduced by applying residue calculus to a contour integral representation of C(s). (See Titchmarsh [8], pp. 18-20.) Several writers have given more elementary proofs of (1) that do not require concepts from advanced real or complex analysis. For example, Titchmarsh [7] obtained a set of complicated recursion formulas which can be used to evaluate C(4), C(6), * *, successively in terms of C(2). Estermann [1] obtained a simpler formula of the same type. These recursion formulas, which show that ((2n) is a rational multiple of C(2)n, were deduced by rearranging absolutely convergent infinite series but did not require any function theory. Estermann also gave an elementary proof of the formula C(2) = ir2/6 as a consequence of Gregory's series -1 +-* * * = 7r/4. 19731 CLASSROOM NOTES 427 A recursion formula simpler than those of Titchmarsh and Estermann was proved by G. T. Williams [11] who showed by elementary methods that I n-1 (4) n + 2 )(2n) = C ((2k)C(2n - 2k). He also obtained the companion result I 1 n (5) (n -2 (1 - 22n)(2n) = z 4(2k - 1)4(2n - 2k + 1), where 0 ()k for s>O. kO (2k + 1) Note that {(1) is Gregory's series for t/4. Taking n = 1 in (5) we find 3((2) = 42 (1) - n2/16, so ((2) = 7r2/6. This result, in conjunction with (4), gives a completely elementary evaluation of ((2n) as a rational multiple of 7r2n. Williams also points out that (4) is equivalent to the following recursion formula for Bernoulli numbers, - (2n + 1)B2n = i (j,)B2kB2n-2k This relation appears in Nielsen's book [5] and was also discovered independently by R. S. Underwood [9] who used it to evaluate the sums 1' I kn in terms of Bernoulli polynomials. The purpose of this note is to show that the elementary method used by Papa- dimitriou to evaluate C(2) in the foregoing paper [6] can be extended to evaluate C(2n) and leads directly to Equation (1) rather than to a recursion formula. The interplay of ideas from elementary algebra and trigonometry makes the proof especially suitable for an elementary calculus course. 2. Elementary Proof of (1). The key ingredient in Papadimitriou's proof is the formula k ki m(2m +1) Ik=1t 2m+1 3 - or rather the asymptotic relation k7r 2 (6) , cot2 =-m2 +O(m) kl1 2m + 1 3 which it implies. Our evaluation of C(2n) makes use of the following lemma which provides a generalization of (6). LEMMA 1. For any integers m > 1, n > 1, we have 428 T. M. A POSTOL [April (7) m 2in _1_ __n- _ _2____2n _2n + o(M2n 1), where the constant implied by the 0-symbol is independent of m. First we show how the lemma implies (1) and then we prove the lemma. The inequality sin x < x < tan x for 0 < x < r/2 implies cot2nx < I < (1 + cot2x)n for each integer n > 1. We take x = k/(2m + 1) and sum on k to obtain M kit (2m k21)2n(m I2 k7rt (8) X cot2n 1 < 2n z kI < + COt2 k= I 2m +.I2fl k1=Ij~ k1 zm +I From (7) and the binomial theorem we see that I 1 + cot 2 k \ -I cott2-k + O(m2n ). k = 1 2m+ I/ k =l 2m + 1 Therefore if we multiply (8) by 7m2n/(2m)2n and let m -+ oo we obtain 1 ni(2ir)2n B2 mr k k2n=(1) 2(2n)! which proves (1). 3. Proof of Lemma 1. As in Papadimitriou's paper we use the polynomial Pm(x) = (2n1 ) - (2m+ I) + (2 j ) - + whose m zeros are the numbers xk =cot2 + k = 1,2, ,m. Let s, = x n + + xn. This sum appears on the left of (7) and we are to prove that 2 4n -1' (9) Sn =2(n -1 _ B2nm2n + O(m2n-1). The proof is by induction on n. The case n = 1 was proved in Papadimitriou's paper. Now we assume that (9) is true for n = 1, 2, **, r - 1, and prove it for n = r, with the help of Newton's formulas (see [10], p. 261) r - 1 (10) - Sr = (- l) rrr + I ( - )rk SkOr-k, r = 1,M2, ,m, k = I 1973] CLASSROOM NOTES 429 where 1, U2, i,m are the elementary symmetric functions of the zeros x x, *, x. In this case we have (2m + 1/ 2m + I 2m(2m-1) (2m-2r + 1) (11) eT~ 2r+1f 1 j (2r +l)! 22r =2r 1)m2r + O(m2r-), for r = 1, 2, *, m. Using this with (9) we find 22r+2k-1B (-1) sk?r_k-(-) =(2k)!(2r + 1 - 2k)! m2r + O(mzrl), so (10) becomes 2r-Ir22r- 1 r-I 22 - sr = 2r-1 m 2r +(1)r-I22r-lM2r Y 2 B2k +o rl (2r + 1)! k=1 (2k)!(2r+ 1- 2k)! +O(m r) = ( r ir,2r-1.I2r ___-r1 2kBk +~ 2r-l1 = (-l)r22-tm2 (2r)! kt=O (2k)!(2r + 1 - 2k)!) Now we use Lemma 2 (stated below) to evaluate the expression in braces and we find 24r-B -Sr = (t 1)r (2r)!2r m2r + O(m2r-1), which proves (9) by induction. 4. A lemma on Bernoulli numbers. LEMMA 2. If r ? 1 we have F 22kB2k _ 1 (12) 2 k =o (2k)!(2r + I - 2k)! (2r)! Proof. Let B,,(x) denote the Bernoulli polynomial defined by (13) B.(x) =S ( )Bsxn-s or, equivalently, by the power series expansion ze- = Bn(X) n ez -1 n I!z, zj2. A well-known property of Bn(x) is the functional equation (14) Bn( - x) = (-1)nBn(x) which follows at once from the identity 430 T. M. APOSTOL [April (1 -x)z ze-xz e' e-ze- ez_1 e~z 1* Equation (14) implies (15) B2r+ (2) = 0. Formula (12) is a disguised form of (15). We use (15) along with (13) and multiply by 22r+1 to obtain 2r+1 (2r+ 1 2SB s0 s In view of (3) this becomes ( 2B3k + k= 2+1 22k B2k = 0, ( t ) k- =( 2k) which is the same as (12). 5. Concluding remarks. When the method of section 2 is applied to evaluate C(2n + 1) we obtain the formula (\2n+ 1 1 M kit (16) 4(2n + 1) ( lim - X cot2 m M kn 1 2m +I or its equivalent, m kir ( 2n+ (17) X Cot2n+' I= I -I 42n + 1)M2+ I o(M2n) as m -+ o. Although (16) expresses C(2n + 1) as a multiple of 7r2n+1 it is not known if this multiple is rational or not. The author has been unable to extend the proof of Lemma 1 to obtain an alternate formula for the asymptotic value (for large m) of the sum in (17). All attempts to estimate this sum lead back to (17). NOTE. After this paper was submitted for publication, a paper appeared by Kenneth S. Williams [12] on the same subject.Williams also uses the cotangent sum of Lemma 1 in his evaluation of C(2n), but his proof, like Euler's, uses complex function theory and cannot be considered elementary. See also I. Skau and E. S. Selmer, Nordisk Mat. Tidskr., 19 (1971) 120-124. References 1. T. Estermann, Elementary evaluation of 4(2k), J. London Math. Soc., 22 (1947) 10-13. 2. L. Euler, De summis serierum reciprocarum, Comment. Acad. Sci. Petropolit., 7 (1734/35), (1740) 123-134; Opera omnia, Ser. 1 Bd. 14, 73-86. Leipzig-Berlin, 1924. 3. Finn Holme, En enkel beregning av J= 11/k2, Nordisk Mat. Tidskr., 18 (1970) 91-92. 4. K. Knopp, Theory and Application of Infinite Series, Hafner, New York, 1951. 5. N. Nielsen, Traite 6l6mentaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923. 1973] MATHEMATICAL EDUCATION 431 6. loannis Papadimitriou, A simple proof of the formula I Ik -2 = 7 2/6, this MONTHLY, 80 (1973) preceding article. 7. E. C. Titchmarsh, A series inversion formula, Proc. London Math. Soc., (2) 26 (1926) 1-11. 8. , The Theory of the Riemann Zeta Function, Oxford, 1951. 9. R. S. Underwood, An expression for the summation imn mP, this MONTHLY, 35 (1928) 424-428. 10. J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948. 11. G. T. Williams, A new method of evaluating g(2n), this MoNrHLY, 60 (1953) 19-25. 12. Kenneth S. Williams, On Yn? 1(1/n2k), Math. Mag., 44 (1971) 273-276. MATHEMATICAL EDUCATION EDITED BY J. G. HARVEY AND M. W. POWNALL Material for this Department should be sent to either of the editors: J. G. Harvey, Department of Mathematics, University of Wisconsin, WI 53706; M. W. Pownall, Department of Mathe- matics, Colgate University, Hamilton, NY 13346. AN INTEGRATED SEQUENCE IN THE MATHEMATICAL SCIENCES FOR UNDERGRADUATE BUSINESS STUDENTS R. H. RANDLES AND A. J. SCHAEFFER, University of Iowa The courses in mathematical science (mathematics, statistics, and computer programming) which are required for every business student vary widely among colleges and universities. In a recent sample survey of midwestern universities, Rodger Collons [1] found that among the 30 schools surveyed on the semester system, the required hours fell between the extremes of 0 and 21. The median of the required hours among those 30 schools was 9. A typical program might therefore consist of one 3 hour course each in mathematics, statistics, and computer program- ming. It is the purpose of this article to describe a sequence of two 4 semester hour courses developed at the University of Iowa in which topics from the three areas of mathematics, statistics and computer programming are blended together in an effort to increase the motivation of each of these subject areas. It is hoped that in so doing, the student will acquire more of an overview of the mathematical sciences and how techniques from all three disciplines lend themselves (possibly in conjunction with one another) to the solution of business problems. This article contains some of the details of this sequence and some suggestions for integrating topics from the mathematical sciences. 1. Course structure. The number of students entering this sequence each year is Article Contents p. 425 p. 426 p. 427 p. 428 p. 429 p. 430 p. 431 Issue Table of Contents The American Mathematical Monthly, Vol. 80, No. 4 (Apr., 1973), pp. 349-474 Front Matter [pp. ] A Unified Theory of Integration [pp. 349-359] Highlights in the History of Spectral Theory [pp. 359-381] The Legend of John Von Neumann [pp. 382-394] Alternating Euler Paths for Packings and Covers [pp. 395-403] Mathematical Notes Stable Laws and the Imbedding of L^p Spaces [pp. 403-407] A Convex Matrix Function [pp. 408-409] Solution of Fejes Tóth's Illumination Problem [pp. 409-410] A Covering Theorem [pp. 410-411] Distributivity over the Dirichlet Product and Completely Multiplicative Arithmetical Functions [pp. 411-414] Perfect Parallelograms [pp. 414-415] A Crowded Set of Non-Intersecting Lines [pp. 415] Research Problems A Deception Game [pp. 416-417] Classroom Notes Traffic Flow: Laplace Transforms [pp. 417-423] Irrational Numbers [pp. 423-424] A Simple Proof of the Formula ∑^∞_{k = 1} = π²/6 [pp. 424-425] Another Elementary Proof of Euler's Formula for ζ(2n) [pp. 425-431] Mathematical Education An Integrated Sequence in the Mathematical Sciences for Undergraduate Business Students [pp. 431-433] Problems and Solutions Elementary Problems: E2408-E2413 [pp. 434-435] Solutions of Elementary Problems E2350 [pp. 435-436] E2351 [pp. 436] E2352 [pp. 436-437] E2353 [pp. 437-438] E2354 [pp. 438-439] E2357 [pp. 439-440] Advanced Problems: 5906-5910 [pp. 440-441] Solutions of Advanced Problems 5832 [pp. 441-442] 5833 [pp. 442-443] 5834 [pp. 443] 5835 [pp. 443-444] 5836 [pp. 444-445] 5837 [pp. 445] 5838 [pp. 445-446] 5839 [pp. 446] Reviews Review: untitled [pp. 447-448] Review: untitled [pp. 448-449] Review: untitled [pp. 449-451] Review: untitled [pp. 451-453] Review: untitled [pp. 453-454] Review: untitled [pp. 454-455] Telegraphic Reviews [pp. 456-465] News and Notices [pp. 466-467] Mathematical Association of America: Official Reports and Communications October Meeting of the North Central Section [pp. 467-468] Officers and Committees as of February 1, 1973 [pp. 468-473] Representatives of the Association [pp. 473] Calendar of Future Meetings [pp. 474] Future Meetings of Other Organizations [pp. 474] Back Matter [pp. ]