The rational sequences Zeta(k,n):=sum(1/m^k,m=1..n), the partial sums of Euler's Zeta(k) series, obtained in the limit n->infinity, are, for k=2..10, and n=1..20, given below. For n=1: Zeta(1,n)=H(n), the harmonic numbers, and the series diverges. ----------------------------------------------------------------------------------------------------- For n>=2 one has (See Abramowitz-Stegun, p.260, 6.4.3.) Zeta(k,n)= (Zeta(m)+((-1)^(m+1))*Psi(m-1,n+1)/(m-1)! with the Zeta(m)=Zeta(1,m) function and the polygamma function Psi(n,x) (the n-th derivative of the logarithmic derivative of the Gamma function. ----------------------------------------------------------------------------------------------------- The g.f.s for the sequences Zeta(k,n), n=1..infty, are G(k;x)= polylog(k,x)/(1-x) with the polylogarithm functions. polylog(k,x) is, per definition, the g.f. for the sequence of the k-th powers of the inverse natural numbers. For the inverse numbers (k=1) this is the polylog(1,x)=-ln(1-x) function, and for k=2 it is the dilogarithm function dilog(1-x) (Maple9). ----------------------------------------------------------------------------------------------------- k=1: Harmonic numbers A001008/A002805. G.f. polylog(1,x)/(1-x)=-ln(1-x)/(1-x). [1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504] Zeta(1) does not exist (is infinite). ------------------------------------------------------------------------------------------------------ k=2: A007406/A007407; Zeta(2)= 1.6449340668... G.f.: polylog(2,x)/(1-x). [1, 5/4, 49/36, 205/144, 5269/3600, 5369/3600, 266681/176400, 1077749/705600, 9778141/6350400, 1968329/1270080, 239437889/153679680, 240505109/153679680, 40799043101/25971865920, 40931552621/25971865920, 205234915681/129859329600, 822968714749/519437318400, 238357395880861/150117385017600, 238820721143261/150117385017600, 86364397717734821/54192375991353600, 17299975731542641/10838475198270720] Values for n=10^m, m=0..5: 1., 1.549767731, 1.634983900, 1.643934568, 1.644834073, 1.644924068 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=3: A007408/A007409; Zeta(3)= 1.2020569031... G.f.: polylog(3,x)/(1-x). [1, 9/8, 251/216, 2035/1728, 256103/216000, 28567/24000, 9822481/8232000, 78708473/65856000, 19148110939/16003008000, 19164113947/16003008000, 25523438671457/21300003648000, 25535765062457/21300003648000, 56123375845866029/46796108014656000, 56140429821090029/46796108014656000, 56154295334575853/46796108014656000, 449325761325072949/374368864117248000, 2207911834254200646437/1839274229408039424000, 245358578943756786493/204363803267559936000, 1683118856778495358491487/1401731326612193601024000, 336658814638864376538323/280346265322438720204800] Values for n=10^m, m=0..5: 1., 1.197531986, 1.202007401, 1.202056403, 1.202056898, 1.202056903 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=4: A007410/A007480; Zeta(4)= 1.0823232337... G.f.: polylog(4,x)/(1-x). [1, 17/16, 1393/1296, 22369/20736, 14001361/12960000, 14011361/12960000, 33654237761/31116960000, 538589354801/497871360000, 43631884298881/40327580160000, 43635917056897/40327580160000, 638913789210188977/590436101122560000, 638942263173398977/590436101122560000, 18249420414596570742097/16863445484161436160000, 18249859383918836502097/16863445484161436160000, 18250192489014819937873/16863445484161436160000, 292007196876357275606593/269815127746582978560000, 24389002905437982598916813953/22535229284522356952309760000, 24389217575736760627770573953/22535229284522356952309760000, 3178449758916874904128641277888913/2936813615588238080381960232960000, 3178468114001972330616643665140369/2936813615588238080381960232960000] Values for n=10^m, m=0..5: 1., 1.082036583, 1.082322905, 1.082323234, 1.082323234, 1.082323234 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=5: A099828/A069052; Zeta(5)= 1.0369277551... G.f.: polylog(5,x)/(1-x). [1, 33/32, 8051/7776, 257875/248832, 806108207/777600000, 268736069/259200000, 4516906311683/4356374400000, 144545256245731/139403980800000, 105375212839937899/101625502003200000, 105376229094957931/101625502003200000, 16971048697474072945481/16366888723117363200000, 16971114472329088045481/16366888723117363200000, 6301272372663207205033976933/6076911214672415134617600000, 6301283671733562325696376933/6076911214672415134617600000, 6301291674250388231757459557/6076911214672415134617600000, 201641519028625175870313858949/194461158869517284307763200000, 286302316744585525852980532588950293/276107037648996202745367733862400000, 95434154288788166027220236648583431/92035679216332067581789244620800000, 236304506035993305443901582534565405715669/227889453271880616207200766816318259200000, 1890436618011579623252753178278440286521/1823115626175044929657606134530546073600] Values for n=10^m, m=0..5: 1., 1.036907341, 1.036927753, 1.036927755, 1.036927755, 1.036927755 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=6: A103345/A103346; Zeta(6)= 1.0173430619... [1, 65/64, 47449/46656, 3037465/2985984, 47463376609/46656000000, 47464376609/46656000000, 5584183099672241/5489031744000000, 357389058474664049/351298031616000000, 260537105518334091721/256096265048064000000, 52107472322919827957/51219253009612800000, 92311616995117182948130877/90738031080962661580800000, 92311647383100199924330877/90738031080962661580800000, 445570781131605573859221176881493/437975145063870303582159667200000, 445570839299219762020391212081493/437975145063870303582159667200000, 2227854388748563035669167509366441/2189875725319351517910798336000000, 142582689233620975717120535871967849/140152046420438497146291093504000000, 3441599639734129841657818562217692625019081/3382929690964537235124904363538251776000000, 3441599739196302352474436585340919249019081/3382929690964537235124904363538251776000000, 161913095162789967079069634260899594958212827455361/159152907672484393983755270823401332001734656000000, 32382619529910829892327658051415140314771727996493/31830581534496878796751054164680266400346931200000] Values for n=10^m, m=0..5: 1., 1.017341512, 1.017343062, 1.017343063, 1.017343063, 1.017343063 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=7: A103347/A103348, Zeta(7)= 1.0083492773... [1, 129/128, 282251/279936, 36130315/35831808, 2822716691183/2799360000000, 940908897061/933120000000, 774879868932307123/768464444160000000, 99184670126682733619/98363448852480000000, 650750755630450535274259/645362587921121280000000, 650750820166709327386387/645362587921121280000000, 12681293156341501091194786541177/12576291107821424895098880000000, 12681293507322704937269896541177/12576291107821424895098880000000, 795732373802519483063688931801166184509/789143616376081512994335288360960000000, 795732381288691429080031515331406184509/789143616376081512994335288360960000000, 795732385907364629635058203181196623933/789143616376081512994335288360960000000, 101853745772435672040195214916143634191549/101010382896138433663274916910202880000000, 41794530981350996538975340812915102041760524474477/41448466476823341693686758239117909939978240000000, 13931510349684302707879332122964873313394948158159/13816155492274447231228919413039303313326080000000, 12452983396984961254768954838142092557598337617274512368501/12349870936173761211842329302824301328031245003653120000000, 2490596681326659584730941156978782465085964605959784505521/2469974187234752242368465860564860265606249000730624000000] Values for n=10^m, m=0..5: 1., 1.008349155, 1.008349277, 1.008349277, 1.008349277, 1.008349277 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=8: A103349/A103350; Zeta(8)= 1.0040773561... [1, 257/256, 1686433/1679616, 431733409/429981696, 168646292872321/167961600000000, 168646392872321/167961600000000, 972213062238348973121/968265199641600000000, 248886558707571775009601/247875891108249600000000, 1632944749460578249437992161/1626313721561225625600000000, 1632944765723715465050248417/1626313721561225625600000000, 350036214341656523811791485040141377/348614789508809898092140953600000000, 350036215152423104696224989140141377/348614789508809898092140953600000000, 285535294511012013599719994461645649102142817/284375793597284734022638664513755545600000000, 285535294703706079490180652561714026702142817/284375793597284734022638664513755545600000000, 285535294814665084460314613710617392218864993/284375793597284734022638664513755545600000000, 73097035489504379781926501380241726163779828833/72800203160904891909795498115521419673600000000, 509907189303751457826768641228113489781192945341106096353/507836558905994020033056646767650020882998991257600000000, 509907189349834795004216817611957628024292411478706096353/507836558905994020033056646767650020882998991257600000000, 8660040895889879602795422202778138097739048470649516007779795689473/8624874212704459432000434464302453009091379853162857470361600000000, 8660040896226788751729190149340655069000863041317148033293969809409/8624874212704459432000434464302453009091379853162857470361600000000] Values for n=10^m, m=0..5: 1., 1.004077346, 1.004077356, 1.004077357, 1.004077357, 1.004077357 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=9: A103351/A103352; Zeta(9)= 1.0020083928... [1, 513/512, 10097891/10077696, 5170139875/5159780352, 10097934603139727/10077696000000000, 373997614931101/373248000000000, 15092153145114981831307/15061903105536000000000, 7727182467755471289426059/7711694390034432000000000, 4106541588424891370931874221019/4098310578334288576512000000000, 4106541592523201949266162797531/4098310578334288576512000000000, 9683010270183857478533136289380321950921/9663601965184210375114147233792000000000, 9683010272056728280376177683851321950921/9663601965184210375114147233792000000000, 102683476368441736434864754827652083087554881272533/102477660980717526752398069144176948412416000000000, 102683476373401681690885212167147843126978881272533/102477660980717526752398069144176948412416000000000, 102683476376067360826287710449789097580152614828757/102477660980717526752398069144176948412416000000000, 52573939905310006815580379830668747845365915138026709/52468562422127373697227811401818597587156992000000000, 6234631892504071525166401712432923053805706595748851067179358373/6222135400490376172009817969792472291863515560646017024000000000, 230912292316127390041346190264217706734259982976273801302939199/230449459277421339704067332214536010809759835579482112000000000, 74512555996593068624603395097996982840739874462671285157977822783324339021/74363205468646504984871372594710987046892058887044655090392629248000000000, 596100447973906474082274762424364477922711354873977969683932655002382097/594905643749172039878970980757687896375136471096357240723141033984000000] Values for n=10^m, m=0..5: 1., 1.002008392, 1.002008393, 1.002008393, 1.002008393, 1.002008393 (Maple9,digits=10). ------------------------------------------------------------------------------------------------------ k=10: A103716/A103717; Zeta(10)= 1.0009945751... [1, 1025/1024, 60526249/60466176, 61978938025/61917364224, 605263128567754849/604661760000000000, 605263138567754849/604661760000000000, 170971856382109814342232401/170801981216778240000000000, 175075181098169912564190119249/174901228765980917760000000000, 10338014371627802833957102351534201/10327742657402407212810240000000000, 413520574906423083987893722912609/413109706296096288512409600000000, 10725658733010631077837980123359391649694009/10715001858996252463926566452828569600000000, 10725658733183684339928277148208512049694009/10715001858996252463926566452828569600000000, 1478623137054473615254787592944842689807988392270632641/1477153996440454717619766727871824205195929190400000000, 1478623137059580374890386255821587524344579342670632641/1477153996440454717619766727871824205195929190400000000, 36965578426553549647808566014281861605597529341720541401/36928849911011367940494168196795605129898229760000000000, 37852752308825227510932683290485196081771414731414031660249/37815142308875640771066028233518699653015787274240000000000, 76310917769836275757123875617429924166816445033269207203147996696551801/76235096239304206549745572122210120713261839892769555622133760000000000, 76310917769857627331413884469115370235288630968406710637654236696551801/76235096239304206549745572122210120713261839892769555622133760000000000, 467867293040677070855607886868901872121523897665303071888471389019703072717616849601/467402426213009930022986138619988554638866745663618654505857352329865461760000000000, 18714691721628908624951710044795227174464940236903680948764104071157286413987212521/18696097048520397200919445544799542185554669826544746180234294093194618470400000000] Values for n=10^m, m=0..5: 1., 1.000994575, 1.000994575, 1.000994577, 1.000994577, 1.000994577 (Maple9,digits=10). ####################################################################################################################################