Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2

Transcription

1 Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the r-th powers of the first atural umbers for positive iteger expoets r. For small values of r expressios for σ r are well ow. For example σ r r + r r σ σ σ We will ot follow Beroulli s origial method. We will use a shortut ivolvig the differetiatio operator. To simplify the presetatio we will osider the sum of the r-th powers till. S r satisfies the differee equatio with iitial oditio S r r + r r r 5 It is easy to derive a expressio for S r for the first few values of r. S S S r + S r r S r S S From this small list we observe that the polyomial S r seems have the followig iterestig properties. The leadig term of S r is r+ r r+. The ext term is. S r 0 S r 0 S r r+ S r + S r has fators,. Whe r is eve, is also a fator. Operator-based Solutio: I the past we have disussed how the Taylor series may be writte as fx + h fx + hf x + h! f x +... fx + h e h fx 3 where is the differetiatio operator. We write S r + S r r as e S r r where the operator meas differetiatio with respet to. A formal solutio is S r e r e r r+ e r + + C where C is a ostat of itegratio to be determied from S r 0 0. Note that the fator was separated so that oly otais positive powers of. Now all that is eeded is a power series expasio for e e i powers of. 5

2 efiitio: e B! The umbers B, 0,,,... are ow as Beroulli umbers. May problems i aalysis a be solved usig Beroulli umbers. Sie /e has simple poles at ±πi,,,..., the expasio here overges for < π. Properties: By lettig 0 i the defiitio we see that B 0 7 Next we see that + e e + e oth 8 is a eve futio of. So i its power series expasio about 0 the odd order oeffiiets are ero. So B +, B 3, B 5, B 7,... are ero. This meas that B 9 B + 0,,, 3,... 0 A reurree formula for the omputatio of the Beroulli umbers will ow be give. e e + e m B +! m! m0 0 B O the left had side there is the produt of two power series. The first is from the defiitio of the Beroulli umbers, ad the seod is the power series for the expoetial futio. We reall the rule for multiplyig two power series. If a b m m 3 the 0 m0 Thus applyig this rule to equatio we get B + B!! 0 a b We ompare oeffiiets of o both sides for > to obtai B B!! Sie B 0 ad B are already ow we oly are about >. B This relatio a be symbolially writte as!! B 0 5 B, > 7 B + B, > 8 where + B is to be expaded just lie a biomial expasio exept that istead of taig supersripts to get powers suh as B we tae subsripts to get the various Beroulli umbers B. Atually sie the B term aels from both sides, we get a relatio ivolvig Beroulli umbers till B. So B 0 + B + B B + B + B B, > 9

3 atually gives B B 0 + B + B B B, > 30 I this formula early half the terms o the right had side do ot otribute aythig sie B 3 B 5 B We show the use of 30 to ompute B ad B. B 3 B 0 + 3B B 3 is ow to be 0. B 5 B 0 + 5B + 0B + 0B Exerise: Usig a suitable arbitrary preisio arithmeti paage write a program to ompute B i fratioal form. Compare your program output with the table give here. B / /30 / 8 /30 0 5/ 9/730 7/ 37/ / / / / / / / / / / / /3530 What may osider to be the first omputer program i the world was writte to ompute the Beroulli umbers by Lady Ada Lovelae85-85 for the Aalytial Egie of Charles Babbage Exerise: Show that What is S 0? Compute S S r r + 3 r r + B r+ 33 Expasios of some futios: From ad the eve futio of i 8 it is lear that Substitutig for we get Substitutig i for we get oth B!, < π 3 oth ot B, < π 35! B, < π 3! 3

4 Sie ot has simple poles at π where a be ay iteger, ot π 37 where the ostats are determied as π lim π ot lim os 38 π π si It should be oted that the sum 37 is uderstood to be the limit as N teds to ifiity of the sum of terms from N to N. So ot + π + π + π 39 ad ot + π π π 0 This expasio is valid at all values of exept oero multiples of π. If we restrit to the iside of the irle of radius π with etre at the origi, the /π < o the right had side. But So ot π u u u + u + u u, u < π, π < π The sum of iverse powers of the atural umbers is the famous eta futio of Riema. Whe the real part of the argumet is greater tha the eta futio a be defied by Now But by 3 ζs s + s + 3 s +... ot ot + Comparig oeffiiets of for,,... i 5 ad we get I partiular, Rs > 3 s ζ π B 5 ζ B π ζ π ζ π 90 ζ π 95 π 8 ζ8 950 Sie for ay atural umber, ζ is positive, shows that B has the same sig as for positive. I other words suessive eve idex Beroulli umbers alterate i sig startig with B. There is o hage i sig from B 0 to B /

5 Exerise: Fid S The Euler-Malauri Summatio Formula Operator erivatio This is a formula relatig sums to itegrals. I aalysis itegrals are frequetly easier to fid tha similar sums ad the Swiss mathematiia L. Euler used this formula first to estimate sums i terms of the itegrals. I diret umerial wor itegrals must be approximated by sums ad the Sottish mathematiia C. Malauri98-7 used this formula to estimate a itegral i terms of a trapeoidal sum. The Euler- Malauri formula is importat ot oly beause of the high auray it provides i umerial wor but also beause of the asymptoti forms whih origiate from it. Stirlig s approximatio for the fatorial a be derived usig the Euler-Malauri summatio formula. Let so that ad Now F x x fξ dξ fx 5 F x fx 5 m F x F m x f m x 53 h fx + h + fx h eh + fx h eh + fx h eh + F x h e h + e h eh F x h e h + e h F x + h F x h oth h F x + h F x B +! h F x + h F x F x + h F x + x+h x+h x fξ dξ fξ dξ + x B h! fξ dξ + B h! F x + h F x B h! f x + h f x f x + h f x Beause the sum o the right had side ivolves differees of ed poit derivatives oly, telesopig a be used to express ay trapeoidal sum i the followig way. h fx + fx + h fx + h + fx + h x+h x fξ dξ + B h! f x + h f x This is the elebrated Euler-Malauri summatio formula. It shows the followig remarable fats. TRAPEZOIAL SUM INTEGRAL + CORRECTION The orretio depeds o derivatives of odd order oly at the ed poits. The orretio ivolves oly eve powers of the step sie h. This fat maes Romberg itegratio a great suess. The sum o the right had side is ofte asymptoti i ature. This formula uses the Beroulli umbers. The derivatio just preseted is ot rigorous ad gives o estimate of the error if oe uses oly a fiite umber terms i the orretio. A rigorous derivatio will be preseted later