An Elementary Proof of a Generalization of Bernoulli s Formula

Transcription

1 Revision: August 7, 009 An Elementary Proof of a Generalization of Bernoulli s Formula Kevin J. McGown Harold R. Pars Department of Matematics Department of Matematics University of California at San Diego Oregon State University La Jolla, CA Corvallis, OR Introduction Te familiar formulas N (N + 1) n and n N (N + 1) (N + 1) 6 are special cases of a general formula discovered by Jaob Bernoulli ( ) and publised postumously in Ars conjectandi (1713). To describe Bernoulli s formula, let a and N be positive integers, ten te sum of te at powers of te first N positive integers is given by (a + 1) n a a ( 1) B N a +1. (1) Te constants B in (1) are te Bernoulli numbers. Here te Bernoulli numbers will be defined by te recurrence B 0 1, ( ) + 1 B i 0, > 0, () i i0 or equivalently, by te power series x e x 1 B! x. (3) (A different convention regarding te Bernoulli numbers is used in [9].) Te first five Bernoulli numbers are B 0 1, B 1 1, B 1 6, B 3 0, B

2 Also we ave B +1 0, for 1,,..., (4) as is seen by verifying tat x/(e x 1)+x/ is an even function and using (3). One reference for (1) (from among many possible references) is []. Bernoulli s original derivation of (1) can be found on pages of [1], te modern translation of Ars conjectandi. See [4] for a proof of te equivalence of () and (3). Some autors (e.g. [], [6]) refer to (1) as Faulaber s formula in onor of Joann Faulaber ( ) wo studied power sums extensively, publising is results in is Academia Algebrae (1631). Te cases of a 1,..., 3 of tis formula were nown to Faulaber, but e did not recognize te pattern in te coefficients tat eventually gave rise to te Bernoulli numbers (see [3], [5], [8]). Altoug te general formula (1) is due to Bernoulli, we will noneteless denote te polynomial on te rigt-and side of (1) by F a (N) in Faulaber s onor. Because of its antiquity, it is believed tat no original volume of Academia Algebrae as ever existed in te Americas. Indeed, te boo is rare in Europe as well. Fortunately, A. W. F. Edwards provided Donald Knut wit a potocopy of te volume (once owned by Jacobi) tat is in te Cambridge University Library. Knut as deposited te annotated potocopy in te Matematical Sciences Library at Stanford University. Wile te title of Academia Algebrae is in Latin, te text is in German. Te style is alien, and te notation, terminology, and typograpy are unfamiliar. Remarably, Knut managed to penetrate tese difficulties and to describe Faulaber s wor in [5]. Also in [5], Knut described te cryptogram in Academia Algebrae tat was solved to demonstrate tat Faulaber new te correct formulas for a and a 3. Additionally, tis wor of Knut sowed tat Faulaber s calculations beyond a 3 were not reliable. It is natural to as weter a formula similar to Bernoulli s formula (1) is valid for non-integral powers of a. Wile instances of approximating formulas for summing non-integral powers of integers ave certainly been publised (for example, see [5] and [7]), it is peraps surprising tat only recently as it been realized tat, wit but modest adjustment, Bernoulli s formula (1) generalizes to complex powers wit real part greater tan 1; in [6] we proved te following result.

3 Teorem 1 Suppose a σ + iτ wit σ 1. If ten lim N m σ + 1 min{ Z : σ < }, γ (m a), F a (N) [ (a + 1) ( 1) B N m +1, ] n a N γ F a (N) (a + 1)ζ( a), (5) were ζ( ) is te Riemann zeta function. Moreover, if a is a non-negative integer, ten te sequence on te left-and side of (5) is constant. A few words of explanation are needed for te case a 1 wic is special: Since te summation on te left-and side of (5) is multiplied by a+1 0, te left-and side of (5) equals N 1 N 1. Te only meaningful interpretation of te rigt-and side of (5) is lim a 1 (a + 1) ζ( a) and tat limit equals 1, because of te fact ζ(s) as a simple pole wit residue 1 at s 1. Notice tat if a is a non-negative integer, ten te teorem says (a + 1) n a N 1 F a (N) + (a + 1)ζ( a), wic is te classical result since (a + 1)ζ( a) ( 1) a B a+1, for all integers a 0, and ence a N 1 F a (N) + (a + 1)ζ( a) ( 1) B N a +1. On te oter and, wen a is non-integral, te polynomial F a (N) as te same formal appearance as te polynomial on te rigt-and side of Bernoulli s formula (1), but it is not equal to any of tose polynomials because te binomial coefficients, defined by (a + 1)a (a + )! 3,

4 are non-integral. Also, wen a is non-integral, a furter refinement of line (5) is possible; namely, (a + 1) n a N γ F a (N) (a + 1)ζ( a) + O ( N β), (6) were β m σ. In [6], Teorem 1, and te refinement (6), were proved using Euler Maclaurin summation and an identity for te Riemann zeta function. For te case in wic a is real, one migt ope tat complex analysis could be avoided and an elementary proof be given. In tis paper, we give suc a proof. To facilitate te elementary proof for real a it is convenient to mae a sligt cange of notation and state te result in te following form: Teorem Suppose a > 1 is real and a m+γ wit m Z, 1 < γ 0. In terms of te ceiling function,, we ave m a. If F a (N) ( 1) B N m +1, ten tere exists a real number C a so tat [ ] (a + 1) n a N γ F a (N) C a. (7) lim N Moreover, if a is a non-negative integer, ten te sequence on te left-and side of (7) is identically zero. Wen a > 1 is non-integral, te value γ from Teorem equals te value γ used in Teorem 1; but, wen a is a non-negative integer, we ave γ γ 1. Wile it would be nice to avoid te differing notations in te two teorems, we feel tat Teorem 1 as a cleaner statement using γ and Teorem yields a cleaner statement using γ. Notice tat, wen a is a non-negative integer, te polynomial F a (N) in Teorem is identical to te polynomial on te rigt-and side of Bernoulli s formula (1) and te factor N γ appearing in (7) is identically equal to 1; in tis case Teorem gives te classical result of Bernoulli. In te final section of tis paper, we give an application to summing te pt roots of te first N positive integers wit an example computation. 4

5 Proof of te Teorem Our aim ere is to give an elementary proof of Teorem. Te idea is simple and is as follows: Wen a is a positive integer, one can prove Bernoulli s formula inductively. In fact, doing so for small integral a is often used to illustrate proof by induction. We will do almost te same inductive process even wen a is non-integral. Of course, te inductive process cannot wor for non-integral a exactly as it does for integral a. Instead, wen a is non-integral tere is an error made at eac step of te induction. We eep trac of te accumulated errors, and teir sum equals te constant C a. Carrying out tis process requires calculation and estimation, but it is noneteless elementary. Te following lemma expresses N γ F a (N) in terms of (N + 1) γ F a (N + 1) and will be te crucial part of our argument. Lemma 3 If a > 1 is real and m, γ, and F a (N) are as above, ten (a + 1)(N + 1) a (N + 1) γ F a (N + 1) + N γ F a (N) { O(N γ 1 ), if a is not an integer, 0, if a is an integer. In order to furter motivate te lemma, we will first demonstrate ow Teorem easily follows. Proof of Teorem. Suppose a > 1 is real. First consider te case in wic a is not an integer. Define te sequence A N (a + 1) n a N γ F a (N), and observe tat A N+1 A N (a + 1)(N + 1) a (N + 1) γ F a (N + 1) + N γ F a (N). (8) By Lemma 3, te rigt-and side of (8) is O(N γ 1 ) and terefore te series (A N+1 A N ) (9) N1 5

6 is convergent. Since te series (9) telescopes, we see tat te sequence A N is convergent as well. Tis proves te teorem wen a is non-integral. In case a is an integer, te argument proceeds as above except tat now te rigt-and side of (8) equals 0. Tus we ave so A M A 1 (a + 1) M 1 N1 (A N+1 A N ) 0, M n a F a (M) A M A 1. To complete te proof, we must verify tat A 1 0. Tis equation is trivial wen a 0. Oterwise, it follows readily from (4) (i.e., te fact tat B 0, wenever 3 is odd) as follows: Using m a, (), and B 1 1, we ave A 1 (a + 1) F a (1) (a + 1) [ m ( 1) ( a + 1 [ m ( ] a + 1 (a + 1) )B (a + 1)B 1 0. )B ] Proof of Lemma 3. First we apply Taylor s teorem to f(t) t p. Te values of p tat we need will be specified later; tey are all positive. Since te t derivative of f is given by we ave f(x 0 + y) f () (t) p (p 1)... (p + 1) t p, q 0 ( ) ( ) p p x p 0 y + (x 0 + ξ) p q 1 y q+1, q + 1 were ξ is between 0 and y. In particular, if we set x 0 N + 1 and y 1, we obtain q ( ) ( ) p p N p ( 1) (N + 1) p + ( 1) q+1 (N + ξ N,p,q ) p q 1, (10) q

7 were ξ N,p,q is between 0 and 1. For convenience, we will simplify (10) by writing ( ) p R p,q (N) ( 1) q+1 (N + ξ N,p,q ) p q 1, q + 1 so tat N p q ( ) p ( 1) (N + 1) p + R p,q (N). (11) 0 Note tat 0, if p is a non-negative integer and p q, R p,q (N) ) O (N p q 1, oterwise. (1) Next we use te Taylor expansion (11) wit q m + 1, p a + 1, for 0, 1,..., m, to express N γ F a (N) in terms of N + 1: N γ F a (N) ( 1) B N a +1 m +1 0 ( ) ( ) a + 1 a + 1 ( 1) + B (N + 1) a +1 + ( 1) B R a +1,m +1 (N) ( 1) B (N + 1) a +1 (13) + + m +1 1 ( ) ( ) a + 1 a + 1 ( 1) + B (N + 1) a +1 (14) ( 1) B R a +1,m +1 (N). (15) 7

8 Note tat line (13) equals (N + 1) γ F a (N + 1). In case a is an integer, we see tat) line (15) equals 0. In case a is not an integer, we ave R p,q (N) O (N p q 1, so tat ) ) R a +1,m +1 (N) O (N a m 1 O (N γ 1 ) olds, for 0, 1,,..., m, and ence line (15) is O (N γ 1. To complete te proof, it suffices to sow tat line (14) equals (a + 1)(N + 1) a. By direct calculation we observe tat ( ) ( ) a + 1 a + 1 ( a ) ( + ). By setting l +, we may rewrite te summation appears below, in te form m+1 l+1 summation, we reorganize it into te form m +1 1 m +1 1, wic and ten, by reversing te order of m+1 l1 l 1. Tis yields: ( ) ( ) a + 1 a + 1 ( 1) + B (N + 1) a +1 m +1 1 ( 1) + ( a )( + ) B (N + 1) a +1 m+1 l1 l 1 ( )( l a + 1 ( 1) l l ) B (N + 1) a l+1 m+1 ( [ a + 1 l 1 ( ] l ( 1) )(N l + 1) )B a l+1 l l1 (16) 8

9 Directly from definition () we ave l 1 ( ) { l 1, l 1, B 0, l > 1, and ence line (16) collapses to (a + 1)(N + 1) a, as desired. 3 An Application Let p be an integer. As an application of Teorem 1, we consider te problem of summing te pt roots of te first N positive integers. Substituting a 1/p in (6) and simplifying, we obtain te identity n 1/p N 1/p p ) + ζ( 1/p) + O ( N (1 p)/p), wic we write informally as n 1/p N 1/p p ) + ζ( 1/p). (17) Using Euler Maclaurin summation to approximate ζ and ζ, one can verify tat ζ(x) [ 1/, 0] olds for x [ 1/, 0] (alternatively, one migt reasonably believe te grap tat any matematics software produces). Tus, we are justified in writing n 1/p N 1/p p ) 1. (18) Indeed, since ζ(0) 1/, te difference between te approximations given by (17) and (18) tends to zero as p. Using p 7 and N 10 6, we perform some example computations using te matematics software SAGE (ttp:// We compute te sum directly using 100 bits of precision (about 30 decimal places), as well as te approximations given by 9

10 (17) and (18). n 1/p seconds, error < 10 3 N 1/p p ) + ζ( 1/p) seconds, error < N 1/p p ) seconds, error < Te direct computation of te sum taes almost 6 minutes, wile te computation of (18) is instantaneous and comes witin of te true value; te computation of (17) taes 0.61 seconds and comes witin of te true value! As a second demonstration of te power of tis metod, we use te approximation (17) to perform a computation tat is infeasible using te sum directly; namely, we set p 7 and N Using 500 bits of precision, we find: n 1/p N 1/p p ) + ζ( 1/p) (19) Te wole computation taes 0.71 seconds! Te error is O(N 6/7 ), so we expect te error in tis approximation to be about ( ) 6/ As a result, it is very liely tat te approximation (19) is correct to all displayed decimal places! All computations were done on a Mac- Boo wit a GHz Intel Core Duo processor and GB of memory, running Mac OS

11 References [1] Bernoulli, Jacob, Te Art of Conjecturing, translated wit an introduction and notes by Edit Dudley Sylla, Baltimore: Te Jons Hopins University Press, 006. [] Conway, Jon H. & Guy, Ricard K., Te Boo of Numbers, New Yor: Springer-Verlag, p. 106, [3] Faulaber, Joann, Academia Algebrae, Darinnien die miraculosisce Inventiones zu den öcsten Cossen weiters continuirt und profitiert werden, Augsburg: Joann Ulric Scönigs, [4] Ireland, Kennet F. & Rosen, Micael I., A Classical Introduction to Modern Number Teory, New Yor: Springer-Verlag, p. 8 30, [5] Knut, Donald E., Joann Faulaber and Sums of Powers, Matematics of Computation, 03 (1993), [6] McGown, Kevin J. & Pars, Harold R., Te Generalization of Faulaber s Formula to Sums of Non-Integral Powers, Journal of Matematical Analysis and Applications, 330 (007), [7] Pars, Harold R., Sums of Non-Integral Powers, Journal of Matematical Analysis and Applications, 97 (004), [8] Scneider, Ivo, Joannes Faulaber , Basel; Boston; Berlin: Biräuser Verlag, 1993, [9] Wittaer, Edmund T. & Watson, George N., A Course of Modern Analysis, 4t ed., Cambridge University Press,