A METHOD FOR THE RAPID NUMERICAL CALCULATION OF PARTIAL SUMS OF GENERALIZED HARMONICAL SERIES WITH PRESCRIBED ACCURACY

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1 UPB c Bull, eres D, Vol 8, No, 00 A METHOD FOR THE RAPD NUMERAL ALULATON OF PARTAL UM OF GENERALZED HARMONAL ERE WTH PRERBED AURAY BERBENTE e roue o etodă ouă etru clculul rd l suelor rţle le serlor roce geerlzte, cu o recze restbltă Folosd tegrăr e tervle lese vtjos se obţ exres sle de clcul cre reduc sure cu ulte orde de ăre De seee, etod erte găsre uor exres sle etru rg le erorlor sufcet de strâse Reducere tulu de clcul fţă de rutele extse e clcultorele ctule este sefctvă Oe rooses ew ethod for the rd clculto of rtl sus of geerlzed hrocl seres wth rescrbed ccurcy By usg tegrto o dvtgeously selected tervls oe obts sle exressos of clculus reducg the suto to servl orders of gtude As well, the gve ethod llows fdg sle exressos for error bouds restrcted tervls The reducto of coutto te s cored to the exstg rutes s sgfct Proble forto Let be the su: ( ), R () For fro () oe obtes the hrocl geerlzed seres tht for > s coverget We cll ( ) N rtl su The roble s to ccurte d rd coute ths su for very lrge, rtculr for coverget seres To ths, oe cosders the fucto: Assst, Det of stee eroutce, Uversty POLTEHNA of Buchrest

2 Berbete f x ( x), x > 0, R, () whch s fte dfferetble The we wrte Tylor forul of the for: f ( x) ( x ) ( x ) ( ) * ( x ) R ( x,, ), N, () where the rest R hs the exresso: ( x ) R ( x,, ) ; ( x) ( x) ( θ) θ x, θ ( 0,); () Further, oe tegrtes the fucto f (x) wth the lts [ -, ], (0;) coveetly chose By deotg ths tegrl, d tkg () to ccout, oe yelds: 5 dx, (, ) x 5 (5) For the rest R, we hve led the e vlue theore O the other hd, by drect tegrto of f(x), oe obts: l, - for ; [( ) ( ) ], for () We wt sle exresso for the su:, The, ccordg to (), oe c get reducto of ters, f there s uber k N, such s: k ; k, (0;) (7)

3 A ethod for the rd uercl clculto of rtl sus 5 Fro (7), oe gets two ossble vlues: / ; (8) Now we wrte the relto (5) for d : ;, ; 5 (), ; 80 (0) By eltg the ter /, oe gets: () Now we tke the su of () fro, to,, to obt:, () were stds for the error ter: 0 ; 05 5 () By deotg the su:, () oe gets the forul for clculto of the rtl su, :, (5)

4 Berbete By usg (), oe yelds: ` ( ) / (( ) ), ( ) /, () the lt, 0, oe gets: () ( ) ( )( ) l l ( ) (-) For 0, oe recover the trvl cse ( 0 ) ( 0, d ) 0 For,, fro () oe obts: ( ) ( ), < 0 ; (-b) (-c) The error evluto ( ) order to fd boud for the error (see ), we cosder, fro () d (0), the ost desdvtgeous cobto of d to get xu ostve vlue: ( ) 5 x ( ) 05, () e the xu dfferece of ters rthess

5 A ethod for the rd uercl clculto of rtl sus 7 Further, we observe tht for y twce dfferetble fucto g(x), oe obts, by usg Tylor forul d tegto the tervl [, ], the exreso: g ( x) dx g g g" ( ), (, ), e () g e beg the e vlue of g(x) the tervl (, ) The, for g (x) > 0, oe obt the egulty: () g e < g ; g" > 0, tht s, for g > 0, the e vlue s sller th the fucto vlue the ddle of the tervl Now, cosder the fucto: 05 x ( x), x, g > x () whch s ostve together wth ts dervtve g (x) for x > / Tkg the tervl [, ], > /, we fd the ddle the vlue rethess fro (): g ( ) 05 (5) fct, the dex tkes lrger vlues ( 7 ) Thus, we get the error evluto: ( ) g ( x) dx g ( x) x 5 ( )( ) ( G ( ) G ( ) ) 0 5 dx ()

6 8 Berbete where the fucto G(x) coes fro tegrto hvg the exresso: G 05 x ( x) x (7) By deotg x the lt for : ( ) ( )( ) 0 05 x ( ), (8) oe gets the bsolute error evluto: Results orsos ( ) ( ) ( ) x (8), x Tbles,, re gve the sus for 05, (hrocl seres) d (coverget seres) desred ccurcy of t lest 5; d 7 exct decl fgures resectvely Tble 05 ( 05) ( 05) Tble exct,x E E E - 7 Exele lculte su 0, wth t lest seve exct decl fgures, by usg Aswer Oe clcultes ( 05) by usg the relto (): ( 05) ;

7 A ethod for the rd uercl clculto of rtl sus the: ( 05) ( 05) ( 05) exct, x E E E - 7 Tble Exle By usg Tble, clculte the su of ters of the hrocl seres foud betwee oe llo d bllo, wth t lest seve exct decl fgures Aswer Oe hs to clculte the dfferece (see ): ( ) ( ) l 0 0 ( 0,exct ( 0 ) ( 0 ) 0,exct ) 0 l 0 0 Rerk ths cse, we hve dfferece error: d oe c exect eve sller errors th tht,0,0 0 reseted Tble By usg Petu couter t 500 MHz, the coutto te s bout 8 utes Tble exct, x E E E - 7 Exle Becuse for, oe gets coverget seres we c clculte the su (see b; c):

8 0 Berbete Tble, the xu error vrto for, ( ), x s reseted: fter cresg o the tervl [0; 075], the xu error s decresg costtly Error vrto wth Tble , x R E F E R E N E Ncolescu M, Dculescu N, Mrcus, Alză tetcă vol, EDP, Bucureşt, tăăşlă O, Alză tetcă, EDP, Bucureşt, 8 Berbete, Mtr, Zcu, Metode uerce de clcul Ed Tehcă, Bucureşt, 77 Berbete, Berbete, Metode ctttve, vol, Ed Bre, Bucureşt, 00