A REMARKABLE FORMULA FOR APPROXIMATING THE SUM OF ALTERNATING SERIES

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1 A REMARKABLE FORMULA FOR APPROXIMATING THE SUM OF ALTERNATING SERIES Thoms J Osler Mthemtics Deprtmet Row Uiversity Glssboro, NJ osler@rowedu Itroductio I this pper we preset simple formul tht c ofte be used to ssist i the pproximtio of the sum of ltertig series () + ( ) f ( ) The ides preseted here were motivted by trsltig Euler s ppers [] d [2] Suppose we begi by summig terms of this series exctly, the the pproximtio tht we will obti is give by (2) 7 f( + ) + f( + 2) 2 f( + 3) ( ) f( ) ( ) f( ) Notice tht we re ddig the simple correctio term (3) 7 f( + ) + f( + 2) 2 f( + 3) A ( + ) 2 to ccout for the ifiite umber of terms tht hve bee eglected It is remrkble tht tht (3) cotis o itegrls or derivtives It is lgebric lier sum of the three cosecutive terms strtig t + This pper is ot rigorous We preset simple iforml derivtio of (3) i the ext sectio, but o estimte of the error is ttempted Rther our purpose is to show

2 2 some remrkble exmples of the use of (3) with series closely relted to the et fuctio The most surprisig exmples occur whe the series diverges 2 Iforml derivtio of the summtio formul Begiig with the Euler-Mcluri summtio formul (see [3]), N b f( b) + f( ) k ( k) k) k f ( + h) f( x) dx ( f ( b) f ( ) ) h 0 h k 2 k! Here b + hn d the umbers B k re the well kow Beroulli umbers The first B2 term i the summtio is ( f '( b) f '( ) ) h, d sice B 2 this term becomes 2! 6 B f '( b) f '( ) h Approximtig the derivtives by 2 f f ( + h) f( ) '( ) d h f f ( b+ h) f( b) '( b) we get h N 0 b f ( b) + f( ) f( ) f( + h) f( b) f( b+ h) f ( + h) f ( x) dx + + h 2 2 2, where we hve dropped terms with powers of h greter th (The bove formul ws first foud by Euler i [] with differet derivtio) Lettig N we get 0 7 f ( ) f( + h) f ( + h) f ( x) dx + h 2 We ow employ ide used by Euler i [2] To obti the ltertig sigs, i plce of h let us write 2h to get the summtio: 0 7 f ( ) f( + 2 h) f( + 2 h) f( x) dx+ 2h 2 Double this d subtrct the precedig series from it to get the ltertig series versio of Euler s summtio formul

3 3 7 f ( ) + f( + h) 2 f( + 2 h) ( ) f( + h) 2 0 Set h d get 7 f( ) + f( + ) 2 f( + 2) ( ) f( ) 2 Notice tht this result is prticulrly simple sice the itegrl hs disppered 3 Exmples of summig ltertig series We will ow give exmples of pproximtig the sum of ltertig series + ( ) f ( ) I ll cses we will begi by fidig the exct sum of the first terms + ( ) f ( ), the dd pproximtio of the remiig terms from (3) which we cll 7 f( + ) + f( + 2) 2 f( + 3) A ( + ) 2 Throughout the pper, umericl d grphicl results were obtied usig Mthemtic Exmple Let f( x) d cosider the series x ( ) + We obti the results show i the followig tble: umber of Exct fiite series Additio of the correctio term terms i ( ) + + ( ) the exct + A ( + ) prt of the sum

4 4 We kow tht this series sums to log , so the lst result i the bove tble is ccurte to 4 deciml plces While 00,000 terms of the exct series yields 5 correct deciml plces, the dditio of the correctio term dditiol 9 deciml plces! A(00,00) dds The remiig exmples re cocered with the et fuctio which is usully defied by the series ζ ( ) This series is vlid for Re( ) > Usig elemetry lgebr it is esy to show tht we c lso use ltertig series η( ) ( ) + to defie the et fuctio s + ( ) ( 2 ) ( 2 ) η( ) ζ ( ) Becuse this series is ltertig, it coverges i the wider rge Re( ) > 0 Exmple 2 I this exmple we exmie the series η (2) ( ) 2 + We fid tht 0000 ( ) ( ) η(2) + A(000) where A(000) is clculted from (3) with f( x) The exct vlue of η (2) is 2 x

5 to 20 deciml plces We see tht the exct fiite sum gives us 8 correct deciml plces, while the dditio of the correctio term (3) dds dditiol 8 correct deciml plces Exmple 3 I this exmple we exmie the Leibi series obti the results show i the followig tble: ( ) + 4 π We 2 umber of terms i the exct prt of the sum Exct fiite series ( ) Additio of the correctio term ( ) A( + ) The vlue of π correct to 20 digits is We see tht the dditio of our three term correctio more th doubles the umber of correct digits 4 Exmples of diverget ltertig series The series + ( ) η ( ) c be used to compute the et fuctio from ( ) ζ( ) 2 η( ) However this series oly coverges i the right hlf ple 0< Re( ) To fid vlues of the et fuctio i Re( ) 0 we usully employ the fuctiol equtio

6 6 s π s ζ ( s) 2(2 π) cos Γ() s ζ() s 2 Surprisigly, the series ( ) + c be used to roughly estimte vlues of the et fuctio eve i the regio where it diverges provided we re ot too fr wy from the imgiry xis We will demostrte this i the followig exmples Agi, we will use the pproximtio + + ( ) ( ) + A ( +, ) where from (3) we hve (4) 7 2 A ( + ) + 2 ( ) + ( + 2) ( + 3) Exmple 4 We estimte η( /2) usig the diverget series our computtios re show i the followig tble: ( ) + The results of /2 umber Exct fiite series Additio of the correctio term of terms ( ) + + ( ) i the /2 + A ( + ) /2 exct prt of the sum The exct vlue is η( /2) to 20 deciml plces I this cse the exct fiite series gives o idictio of the correct umericl vlue Yet with the

7 7 dditio of the correctio term (4) we were ble to get 8 correct deciml plces from this diverget series This looks to us like mircle! Exmple 5 We c lso clculte with complex vlues We estimte η( /2 + i) usig the diverget series + ( ) The results re show i the followig tble: /2 +i umber of terms i the exct prt of the sum Exct fiite series ( ) + /2+ i Additio of the correctio term ( ) + /2+ i + A ( + ) i i i i i i i i The exct vlue is η ( / 2 + i) i to 20 deciml plces As i the previous exmple, the exct fiite sum gives o idictio of the correct umericl vlue, yet 5 correct deciml plces were obtied by the dditio of the correctio term (4) Exmple 6 We estimte η( 3/2) usig the diverget series re show i the followig tble: ( ) + The results 3/2 umber of terms i the exct Exct fiite series Additio of the correctio term

8 8 prt of the sum ( ) 3/2 + ( ) + 3/2 + A ( + ) The exct vlue is η (3 / 2) to 20 deciml plces Appretly we re er the limits of this method Exmple 7 I this fil exmple we cosider usig the diverget series ( ) + y i to estimte the vlue of η ( + iy) for 0 y 50 I figure we see y plotted s the horiotl xis with the sum of the first 00 terms of the series plotted verticlly Figure : The sum of 00 terms of the exct series ( ) I Figure 2 we dd the correctio term A(0) to the series show i the previous figure We compre this with η( + yi) 00 + yi

9 9 Figure 2: Comprig the exct vlue of η ( + iy) with the corrected series 00 ( ) + yi + A(0) The grphs show tht the dditio of the simple correctio term (0) A + + yi + yi + yi yields vlues tht re surprisigly close to η ( + iy) Refereces [] Euler, L, Methodus uiverslis serierum covergetium summs qum proxime iveiedi, (A geerl method for fidig pproximtios to the sums of coverget series (E46)), Origilly published i Commetrii cdemie scietirum Petropolite 8, 74, pp 3-9 Oper Omi: Series, Volume 4, pp 0 07 A Trsltio with Notes by W Jcob d T J Osler, s well s the origil pper re vilble o the web t the Euler Archive,

10 0 [2] Euler, L, Remrques sur u beu rpport etre les series des puissces tt directes que reciproques (Remrks o beutiful reltio betwee direct s well s reciprocl power series (E254)), Origilly published i Memoires de l'cdemie des scieces de Berli 7, 768, pp Oper Omi: Series, Volume 5, pp A Trsltio with Notes by L Willis d T J Osler is vilble o the web t the Euler Archive, [3] Kopp, Kord, Theory d Applictio of Ifiite Series, Dover Publictios, New York, 990 (A trsltio by R C H Youg of the 4 th Germ dditio of 947) ISBN: