Numerele lui Bernoulli

Numerele lui Bernoulli sunt termenii unui șir, $B_n, \!$ de numere raţionale, definit prin relaţia:

$\frac{x}{e^x-1} \equiv \sum_{n=0}^{\infty} \frac{B_n x^n}{n!}. \!$

Sunt atribuite lui Jacques Bernoulli.

Şirul numerelor lui Bernoulli are aplicaţii la studiul funcţiilor trigonometrice, teoria numerelor şi în analiză matematică.

Primele 100 de numere Bernoulli sunt:

Bernoulli(0) 1

Bernoulli(1) -1/2

Bernoulli(2) 1/6

Bernoulli(4) -1/30

Bernoulli(6) 1/42

Bernoulli(8) -1/30

Bernoulli(10) 5/66

Bernoulli(12) -691/2730

Bernoulli(14) 7/6

Bernoulli(16) -3617/510

Bernoulli(18) 43867/798

Bernoulli(20) -174611/330

Bernoulli(22) 854513/138

Bernoulli(24) -236364091/2730

Bernoulli(26) 8553103/6

Bernoulli(28) -23749461029/870

Bernoulli(30) 8615841276005/14322

Bernoulli(32) -7709321041217/510

Bernoulli(34) 2577687858367/6

Bernoulli(36) -26315271553053477373/1919190

Bernoulli(38) 2929993913841559/6

Bernoulli(40) -261082718496449122051/13530

Bernoulli(42) 1520097643918070802691/1806

Bernoulli(44) -27833269579301024235023/690

Bernoulli(46) 596451111593912163277961/282

Bernoulli(48) -5609403368997817686249127547/46410

Bernoulli(50) 495057205241079648212477525/66

Bernoulli(52) -801165718135489957347924991853/1590

Bernoulli(54) 29149963634884862421418123812691/798

Bernoulli(56) -2479392929313226753685415739663229/870

Bernoulli(58) 84483613348880041862046775994036021/354

Bernoulli(60) -1215233140483755572040304994079820246041491/56786730

Bernoulli(62) 12300585434086858541953039857403386151/6

Bernoulli(64) -106783830147866529886385444979142647942017/510

Bernoulli(66) 1472600022126335654051619428551932342241899101/64722

Bernoulli(68) -78773130858718728141909149208474606244347001/30

Bernoulli(70) 1505381347333367003803076567377857208511438160235/4686

Bernoulli(72) -5827954961669944110438277244641067365282488301844260429/140100870

Bernoulli(74) 34152417289221168014330073731472635186688307783087/6

Bernoulli(76) -24655088825935372707687196040585199904365267828865801/30

Bernoulli(78) 414846365575400828295179035549542073492199375372400483487/3318

Bernoulli(80) -4603784299479457646935574969019046849794257872751288919656867/230010

Bernoulli(82) 1677014149185145836823154509786269900207736027570253414881613/498

Bernoulli(84) -2024576195935290360231131160111731009989917391198090877281083932477/3404310

Bernoulli(86) 660714619417678653573847847426261496277830686653388931761996983/6

Bernoulli(88) -1311426488674017507995511424019311843345750275572028644296919890574047/61410

Bernoulli(90) 1179057279021082799884123351249215083775254949669647116231545215727922535/ 272118

Bernoulli(92) -1295585948207537527989427828538576749659341483719435143023316326829946247/1410

Bernoulli(94) 1220813806579744469607301679413201203958508415202696621436215105284649447/6

Bernoulli(96) -211600449597266513097597728109824233673043954389060234150638733420050668349987 259/4501770

Bernoulli(98) 67908260672905495624051117546403605607342195728504487509073961249992947058239/6

Bernoulli(100) -945980378191221252952274330694937218727028415330669361333856962043113954151972 47711/33330

Vezi şi

• Numerele prime ale lui Fermat

Resurse

• The Bernoulli Number Page
• Wolfram MathWorld
• Prime divisors of the Bernoulli and Euler Numbers
• The first 498 Bernoulli Numbers
• Potenzsummen, Bernoulli-Zahlen und Euler'sche Summnenformel
• Polynôme de Bernoulli - Nombres de Bernoulli - Applications
• Autour des nombres et des polynômes de Bernoulli
• Nombres et polynômes de Bernoulli
• À la découverte des nombres de Bernoulli

Familia Bernoulli
Jacques Bernoulli (Jakob Bernoulli) (1654 - 1705) Ecuația diferențială de tip Bernoulli Numerele lui Bernoulli Lemniscata lui Bernoulli Operatorul Bernoulli Inegalitatea lui Bernoulli	-frate-	Jean Bernoulli (Johann Bernoulli) (1667 – 1748) Identitatea lui Bernoulli Regula lui Bernoulli
		| fiu |
		Daniel Bernoulli (1700–1782) Legea lui Bernoulli Teoria cinetică a gazelor Teoria probabilităților