AMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.

Transcription

1 J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order. he recursio is obtaied by expadig i Fourier series certai elemetary fuctios. AMS Mathematics Subject Classificatio : 40A05, 40A99, 4A0. Key words ad phrases : Harmoic series, Fourier series.. Itroductio It is a classical result that ) k ) k + ) k = δ k, k N. ) he δ k s i the formula above are ratioal umbers which are give i terms of the Beroulli umbers B k or the Euler umbers E k )/ depedig o whether k is eve or odd. he treatmet of the case whe k is eve ca be traced back to Euler, who was the first to show, for example, that for k =, δ k = /3, cf. [5], pp here are several other works dealig with this case; see e.g. [], [3], [6], [7]. he case k odd, o the other had, has received less attetio over the years. A elemetary proof give i [] shows that the δ k s ca be obtaied as the volumes of certai covex polytopes, while simultaeously coverig both parity cases referrig to k i ). I this case, i order to fid the sums of, say, the alteratig harmoic series of odd order which correspod to k odd i ) oe caot avoid goig through the computatio of the Euler umbers. A differet approach that relates the computatio of the series to the eumeratio of alteratig ad cyclically alteratig permutatios is give i [4]. he purpose of this ote is to give a coveiet recursive formula for the sums of alteratig harmoic series of odd order. I particular, we also recover ) i Received,. Revised,. c 00x Korea Society for Computatioal & Applied Mathematics.

2 Árpád Béyi the case whe k is odd. Our proof employs some basic facts about Fourier series expasios of periodic fuctios. Although the idea of usig Fourier techiques to deal with harmoic series is ot ew for example, a similar approach was pursued i [7] to deal with the case k eve), we could ot fid the recursio we are about to prove elsewhere i the literature. Furthermore, it is of iterest to ote that the recursive relatio is useful i fidig the sums of the alteratig harmoic series of odd order without the iterveig parameters give by the Euler umbers.. he Recursio Let us deote by S k = Our mai result is the followig ) + ) k+. Propositio. For all k 0, k k+ k + )! = ) j S j k j + )!. Before we give the proof of Propositio, let us first recall a few basic facts about Fourier series. Let f : R R be a periodic fuctio of period > 0, that is fx + ) = fx) for all x. I this case, to study f is eough to restrict our attetio to ay iterval of legth, sice the values of f o this iterval determie precisely the fuctio o the whole real lie. We ca also attach to f its Fourier series, fx) a 0 + = a cos x + b si x ), ) where a 0, a, ad b,, are called the Fourier coefficiets of the fuctio f ad are give by a = / / fx) cos x dx ad b = / / fx) si x It is easy to see that if f is, additioally, a odd fuctio, that is fx) = f x) for all x, the a = 0, 0 ad b = 4 / fx) si x dx,. It is 0 importat to ote that, i geeral, oe caot simply replace the symbol with a equality i ). However, whe f behaves icely, this is possible. I dx. 3)

3 Alteratig Harmoic Series 3 particular, if f is a -periodic fuctio so that i, ) the derivative f x) exists ad is fiite, the oe ca write fx) = a 0 + a cos x + b si x), 4) = meaig that the Fourier series attached to f is coverget to fx). his well kow fact follows immediately, for example, from the so called Dirichlet-Jorda theorem o the covergece of Fourier series; see [8], pp , for further details. Cosider ow the fuctios f k : [0, ), f k x) = x k+, k 0. For each k, we will also deote by f k the odd -periodic extesio of f k to the whole real lie, i.e., f k satisfies f k x) = f k x) ad f k x + ) = f k x) for all x. Obviously, the fuctios f k are differetiable o, ) ad all the -traslates of this iterval) ad are discotiuous at all the poits which are odd iteger multiples of. As we have remarked above, sice the f k s are odd fuctios, we ca write for x, ) f k x) = b,k si x, where b,k = = 0 f k x) si x dx. 5) With these prelimiaries, we are ow ready to proceed with the proof of our mai result. Proof. Itegratig twice by parts i the itegral above which defies b,k we obtai b,k = )+ k kk + ) b,k. 6) Let α,k = )+ k kk + ) ad β,k =. he 6) ca be rewritte as b,k = α,k + β,k b,k. 7) Fix ad write ow the correspodig relatios for b,k, b,k,..., b, i 7). If we elimiate all the terms b,j, j k, from the system of equatios obtaied this way, we fid that k b,k = α,k + α,k j β,k β,k β,k j. 8)

4 4 Árpád Béyi Sice β,k β,k β,k j = ) j+ k + )!, from 8) we have k j )! b,k = )+ k + )+ k + )! k k ) j+ k j )!. 9) If we ow let x = i 5) we obtai ) ) f k = b,k si = ) b +,k. = Usig the latter formula ad 9), ad after dividig out both sides by k, we get k+ = ) + +k+)! ) k ) j+ + + ) k j )! or k+ = ) k + + k + )! ) j+ k j )! ) + ) j+3. If we ow chage the idex of summatio i the secod term accordig to j := j, the last equatio becomes k k+ = k + )! ) j k j + )! which is the recursio we wated to prove. ) + ) j+, I particular, if we let k = 0 i our recursive relatio we recover Leibiz s formula 4 = Similarly, we fid 3 3 = = It is also clear that the recursio give i Propositio proves ) for k odd by a easy iductio argumet.

5 Alteratig Harmoic Series 5 Refereces [] Á. Béyi, Fidig the Sums of Harmoic Series of Eve Order, College Math. J., to appear. [] F. Beukers, J. Kolk, ad E. Calabi, Sums of Geeralized Harmoic Series ad Volumes, Nieuw Arch. Wisk. 993), 7-4. [3] H.. Davis, he Summatio of Series, Pricipia Press of riity Uiversity, 96. [4] N.F. Elkies, O the Sums k= 4k +), Amer. Math. Mothly 0 003), [5] V.J. Katz, A History of Mathematics: A Itroductio, Addiso-Wesley, 998. [6] E. Popovici, G. Costovici, ad C. Popovici, he Calculatio of Sums of Harmoic Series of Eve Power, Bul. Ist. Politehic Iaşi, Sect ), 9-. [7] I. Sog, A Recursive Formula for Eve Order Harmoic Series, J. Comp. Appl. Math. 988), [8] A. Zygmud, rigoometric Series I ad II, d ed., Cambridge Uiversity Press, 993. Árpád Béyi received his BA from West Uiversity of imişoara Romaia) ad PhD from Uiversity of Kasas uder the directio of Rodolfo H. orres. He is curretly a Visitig Assistat Professor at Uiversity of Massachusetts, Amherst. His research iterests are i harmoic aalysis ad its applicatios i partial differetial equatios. He has also published several papers o elemetary mathematics. Departmet of Mathematics ad Statistics, Uiversity of Massachusetts, Amherst, MA , USA beyi@math.umass.edu