Sums of generalized harmonic series and volumes

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1 Sums o generalized harmonic series and umes by Fris eukers, Eugenio Calabi and Johan A.C. Kolk Mahemaisch Insiuu, Rijksuniversiei Urech PO ox 8, 358 TA Urech, The Neherlands Deparmen o Mahemaics, Universiy o Pennsylvania Philadelphia, PA -385, USA Inroducion. I is well-known ha he Riemann zea-uncion as deined by Euler): R ) akes elemenary values i.e., values deinable in a irs-year calculus class) when is a posiive even ineger more precisely:! Here, denoes he -h ernoulli number we use he convenion whereby he raional numbers are deined rom he Maclaurin series coeiciens o he ollowing uncion: $&% Equivalenly, he series ) can be replaced by one wih he same erms resriced o odd values o, yielding: whence: -+ ' ') *,+.!! N ) N 3) I we replace he series o eiher ) or / ) or 3) wih anoher o he same erms in absolue value bu alernaing in sign, hen here is no similar elemenary ormula or he sum. However, or odd posiive ineger values o here is he similar ormula due o Euler: where he Euler numbers, or,+ 3 sec cos! ) N, are he naural numbers deined according o: ' ')! 5)

2 I? Among he convenional proos o 3) and ), mos o hem inve eiher residue heory o complex uncions, or Miag-Leler expansions, or echniques rom Fourier analysis. This means ha he average undergraduae majoring in mahemaics could rarely learn i beore he senior year unless she/he is unusually alened). The aim o his paper is o presen a simulaneous proo o 3) and ) in he join ormulaion: -+ 3! N 6) Here each is a posiive raional number, arising as he ume o a cerain -dimensional convex polyope 5 wih raional verices. y direc compuaion o 5 we ind: sec + an ' ')! On expanding he righ side o 7) we obain he expressions or he in erms o he ernoulli and Euler numbers as indicaed above. The proo requires no special knowledge beyond wha is included in a good second-year calculus class, and i is in our seps. Sep I: using sandard echniques on convergence we conver he sum o he series ino an inegral o a raional uncion over he uni cube in R. Sep II: by means o a nonrivial subsiuion o variables, and his is he hear o he maer, he inegral is shown o be equal o 76, wih as above. Sep III: by an analysis o some elemenary inequaliies, we dissec 5 -dimensional ume o he basis o such a pyramid. Sep IV: he ume o his basis is deermined by means o a wo-sep recursion. This requires mahemaical inducion wih respec o. ino congruen pyramids and we express in erms o he From series o inegral over a cube. Lemma. The ininie series 6) converges or each We begin wih he ollowing : = > < < < Here? is he open uni cube R saisying AC denoes -dimensional inegraion over? wih respec o variables G. 7) N, and is sum is represened by he inegral:, or DFEGDF, while. Proo: Firs noe ha he deiniion o he inegral needs some care in he case o even, as he inegrand becomes ininie a he corner o he closure o?. Since he inegrand is posiive in he open se?, in order o prove convergence o he inegral i suices o show he exisence o an increasing sequence o ses HJILKF? such ha M N I N HJI and ha he O*P = QR S QRT U U U T S exis and converge as VXW. We shall do his, hereby proving our lemma, by choosing he subses H I o be he conraced hypercubes V*? wih V. Noe ha: 8 Y = G 8 V < < < Y = > V < < < Expansion o he righ side ino a geomeric series yields: 8 V Q S < < < = Z 8 < < < 8) =

3 I p T E E? 6 For ixed V and summaion o obain:, he series converges uniormly or [? V The inegraion is easy now in view o: The inal resul is: 8 \] < < < Q S = ^ 8 \ = < < < < < <, allowing us o inerchange inegraion = Z = -+ 8 Y = G ] V Q S < < < -+ The series converges uniormly or V [_ ` and hence we can ake he limi or VXW, inding he series o he lemma. u his implies ha he limi o he inegrals exiss or VXW, and ha i is equal o he inegral o he lemma. This yields he desired equaliy. a From cube o polyope. I he inegral in 8) is immediaely seen o be equal o, hence b. So we assume dc rom now on. In order o evaluae he inegral 8) his is by no means elemenary), we make a surprising change o variables. In wha ollows we shall regard he indices E o he coordinaes o a poin in R as inegers modulo, so we compue wih hem cyclically. The subsiuion o variables is given by he ollowing equaions wih he convenions on indices jus described: e sin cos N mod ) Lemma. The change o variables represened by ) gives a diereniably inverible one-o-one or! and he se 5 or. Here 5 is he open -dimensional correspondence beween he se? convex polyope consising o he [ R saisying: + The Jacobian deerminan g gh N mod o he correspondence equals Y < < < Noice ha he Jacobian is exacly he denominaor o he inegrand in 8) his is o course he raison d êre o he subsiuion). Proo: We denoe he correspondence by ikj!! cyclically) and hus:! 5 ml. For! 5 sin sin! ) cos Hence Y, and using his we ge i n?. Conversely, given o? uncion pqj R l R given by: 3 o < < < r < < <., we have ) /, consider he linear

4 p T E 5 Since ' p s 'u. < < < ' ', he mapping p sends ] [ ino isel conracively, and hence p has a unique ixed poin?. Now selec he unique elemen! 5 saisying: v sin sin ev sin cwe-c Then i sin, and his ollows rom p sin. Tha is, he equaion i xy has a unique soluion! 5. This gives ha i is diereniable, one-one and ono. The Jacobian marix zgi has he ollowing simple orm, where we use and as an abbreviaion or sin and cos respecively: comes down o sin Consequenly he Jacobian g Y The diereniabiliy o i gh v < < < 3 3 < < < 3 < < < de z^i ac ha he Jacobian is posiive on! 5 equals: Y : u < < < sin ollows rom he inverse uncion heorem see [MT], p.88) and he. a Noe ha he equaions ) can be solved explicily one obains: } arcsin + ~ ~ x Y ] < < < N mod Now ransorm he inegral 8) by he subsiuion ), and deduce rom Lemma : 8 : < < < = ^! =!! Combining his resul wih Lemma we ind he ollowing Theorem 3. The ormula 6) above holds wih equal o he ume o he 5 o Lemma.

5 ~ b 5 = i j a Dissecing he polyope ino congruen pyramids. The cyclic permuaion o he coordinaes in R deines an orhogonal linear ransormaion o R. The ransormaion preserves he inequaliies ) deining 5, and hereore 5 is mapped isomerically ono isel by. Now denoe by he collecion o 3 5 saisying. or DAEƒDA. In addiion, deine he ses ~, or D similarly, bu wih he index inerchanged wih. Since ~, all hese ses have equal ume and are pairwise disjunc, and he closure o 5 is he union o he closures o he The nex lemma shows ha basis, i.e., ~ s. Hence c. Theorem 3): Noe ha is characerized by he inequaliies: q D Ê D + DwE-Dw Indeed, he wo missing equaions now are a consequence o he given ones, since + + and + q +. Le be he inersecion o he closure wih he coordinae hyperplane i and only i: Š D Ê D + DwE-Dw Lemma. iœj> is he pyramid in R wih is he convex hull o and. ] [ l given by inverible one-o-one correspondence, and is Jacobian is given by > Ž > ) ). Obviously 3) ) as is apex and as is + is a diereniably. Proo: Suppose belongs o he pyramid im i. Then Y b F + b Fu, since. Furhermore + implies u + u + + +, or DwE-Dw. According o ) his means [. Conversely, assume [. Then G i > is solved by and u. Firs noe b, or every hence. The inequaliy implies, while + gives +, or D EJD@. In view o 3) we have, and so im i. The compuaion o he Jacobian is immediae. As a consequence o he lemma we ind: Combining ) and ) we see: = ) 5)

6 % + + Two-sep recursion or he ume o he basis o he pyramid. only i q, q r,, q T < < < From 3) we see o i and r. Hence: T = = = < < < In order o compue his inegral we inroduce polynomial uncions š on R by: š š *T š = œ hen š Now we have, or R: š š = T š = š s T š = = Hence by mahemaical inducion on we ind, or [ R: where š s T š ] š o Ÿ ) ž ž Ÿ E or i is even or odd, respecively. Since š, or E E š = N, we obain: N 6) Noice ha 6) gives an algorihm or recursive compuaion o he, and hence also o he. The le hand sides in 6) occur as he coeicien o, and respecively, i one perorms he muliplicaion in he le hand side o he ollowing ideniy o ormal power series and his implies: + Recognizing he Maclaurin expansion or cos and sin we ind in view o 5) he resul o 7): x cos + an sec + an ' '! Recall he expansion 5) or sec. The uncion an is an odd uncion whose odd-order derivaives a belong o N, and i has he power series an, where he are given by see [C], p.3 or a shor proo):. u In [K] one inds ables or he Euler and ernoulli numbers. Insering boh hese expansions in 7) we obain he ollowing equaliy o power series, which relaes he umina o he analyically deined numbers o 5) and o ): + ' '! I we combine his resul wih 6) we obain 3) and ). Wih his we eel we have compleed our shor circular excursion hrough some analysis, some geomery, some combinaorics and back o he ried and amiliar grounds o elemenary subjecs.

7 Acknowledgemen. The auhors are graeul o D. Zagier or acing as a rai d union beween hem, and o T.A. Springer or his ineres during he preparaion o his manuscrip. Reerences [C] Couran, R. - Dierenial and Inegral Calculus. Volume I. Second Ediion. Wiley Classics Library Ediion. Inerscience Publishers: 88 [K] Knuh, D.E., uckholz, T.J. - Compuaion o angen, Euler and ernoulli numbers. Mah. Comp., ) [MT] Marsden, J.E., Tromba, A.J. - Vecor Calculus. Third Ediion. W.H. Freeman and Company: New York 88

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural