ACCELERATING CONVERGENCE OF SERIES

Transcription

1 ACCELERATIG COVERGECE OF SERIES KEITH CORAD. Itroductio A ifiite series is the limit of its partial sums. However, it may take a large umber of terms to get eve a few correct digits for the series from its partial sums. For example,. coverges but the partial sums s + /4 + /9 + + / take a log time to settle dow, as the table below illustrates, where s is trucated to 8 digits after the decimal poit. The 000th partial sum s 000 wids up matchig the full series. oly i s That the partial sums s coverge slowly is related to the error boud from the itegral test: s + r where. r + < dx x. To approximate. by s correctly to 3 digits after the decimal poit meas r <.000 /0 4, so the boud i. suggests we make / /0 4, so I the era before electroic computers, computig the 000th partial sum of. was ot feasible. Our theme is speedig up covergece of a series S a. This meas rewritig S i a ew way, say S a, so that the ew tail r > a goes to 0 faster tha the old tail r > a. Such techiques are called series acceleratio methods. For istace, we will accelerate. twice so the 0th accelerated partial sum s 0 is more accurate tha the 000th stadard partial sum s 000 above. Let. Series with positive terms: Kummer s trasformatio a be a coverget series whose terms a are positive. If {b } is a sequece growig at the same rate as {a }, meaig a b as, the the limit compariso test. If we happe to kow the exact value of B. a b + a b B + b coverges by b a a b, the The lower boud r > + dx/x / + proves r < > 0000, so 0000.

2 KEITH CORAD ad the series o the right i. is likely to coverge more rapidly tha the series o the left sice its terms ted to 0 more quickly tha a o accout of the ew factor b /a, which teds to 0. The idetity. goes back to Kummer [5] ad is called Kummer s trasformatio. Example.. We will use. to rewrite. as a ew series where the remaider for the th partial sum decays faster tha the error boud / i.. A series whose terms grow at the same rate as. is, which has exact + value B from the simplest example of a telescopig series:. as. Takig a ad b , so b a,. says Lettig s +, here are its values trucated to 8 digits after the decimal + poit for the same as i the previous table. This seems to coverge faster tha s s We have s + r where r.4 r < This teds to 0 faster tha : 3 < dx x 3. Therefore r <.000 if /.000, which is equivalet to 7, ad that s a great improvemet o the boud 0000 to make r <.000. Sice s , the series. lies betwee s ad s , ad sice / whe 000, the value of s 000 tells us. is to 5 digits after the decimal poit. By acceleratig. eve further we ll approximate it to 5 digits usig a far earlier partial sum tha the 000-th. From the series o the right i.3, let a +. A sequece that grows at the same rate as a is b + +, ad we ca compute b exactly usig a telescopig series: as,.5 b / + / /

3 I. with a ACCELERATIG COVERGECE OF SERIES 3 + ad b Feedig this ito the right side of.3,.6 Whe s previous tables for s ad s , we have B 4 ad b a :, the ext table exhibits faster covergece tha + + for the same values of s Lettig r + + +, so s + r, we have r < + 4 < x 4 dx 3 3, which improves o.4 by a extra power of just as.4 improved o. by a extra power of. We have r <.000 if /3 3 <.000, which is equivalet to 9, so from the value of s 0 i the table above,. is betwee s ad s : the series. is.644 to 3 digits after the decimal poit. Let s accelerate the series o the right i.6: for a + +, a sequece growig at the same rate that is exactly summable is b , where /3.7 B b + + / ad b a, so. tells us Feedig this ito.6, Settig s , we have the followig values

4 4 KEITH CORAD s We have s + r where r r < < 6 x 5 dx , has the boud so r 5 < Usig the table above,. is betwee s > ad s < , so. is.6449 to 4 digits after the decimal poit. We ca cotiue this process. For each k, telescopig series like.,.5, ad.7 geeralize to /k + + k + + k /k k.9 k k! ad this lets us geeralize.3,.6, ad.8 to k.0 j + k! k j for each k 0, where the first sum o the right is 0 at k 0. The remaider term r k for the th partial sum of the rightmost series i.0 satisfies. r k < Put k 5 i.0 ad let s 5 5 We get the followig values. k! k!/k + dx xk+ k s By., / s 5 0 r5 0 < 0/6/06 / , which puts. betwee s ad s The series. that we have bee fidig good approximatios to has a exact formula: π This beautiful ad uexpected result was discovered by Euler i 735, 6 whe he was still i his 0s, ad it is what first made him famous. Before fidig the exact value π /6, Euler created a acceleratio method i 73 to estimate. to 6 digits after the decimal poit, which was far beyod feasible had calculatios usig the terms i.. Figure shows Euler s estimate o the secod lie, take from the ed of his article. A accout of this work is i [6], ad the origial paper ad a Eglish traslatio is []. Figure. Ed of Euler s article where / is estimated as

5 ACCELERATIG COVERGECE OF SERIES 5 Example.. Cosider. Ulike., there is o kow formula for this series i 3 terms of more familiar umbers. We will estimate the series by acceleratig it four times. The th term a 3 grows at the same rate as b, ad we kow + + the exact value of b : by.5, it is, so by b a a ow let a , so a grows like rate whose exact sum is kow is b b so by. ad algebra A sequece growig at the same 4 3 : by.9, ! 6, b a a ext let a 3, which grows like. A sequece growig at the same rate whose exact sum is kow is b : by.9, b so by. ad algebra ! 96, b a a It is left to the reader to derive the ext acceleratio, which is We ow have five partial sums that each ted to s 3, s , as : 3 s s , , s

6 6 KEITH CORAD The table below compares these partial sums for several values of, each partial sum beig trucated ot rouded to 8 digits after the decimal poit s s s s s We ca boud the remaider term for each partial sum usig the itegral test, as i our previous example: r : 3 < dx x 3, ad r : r : r : + + r 4 : < < < < < < < 74 7 < 3 x 4 dx 3, dx x5 4 4, dx x , dx x These bouds imply r <.0000 for 4, r <.0000 for 47, r <.0000 for 3, r <.0000 for 6, ad r4 <.0000 for 3. Usig the bouds o r, lies betwee s ad s We also have r 4 < for 9, so lies betwee s ad s Aalogous to the k-fold acceleratio.0 for 3 k j is a k-fold acceleratio of 3 : c j j + j +! + c k+ + k +! k + for each k 0, where the first sum o the right is 0 at k 0 ad c k, 3,, 50, 74,... is determied by the recursive relatio c ad c k kc k + k! for k. The itegers c k are the usiged Stirlig umbers of the first kid that cout the umber of permutatios of the set {,..., k + } havig disjoit cycles.

7 The Leibiz series ACCELERATIG COVERGECE OF SERIES 7 3. Alteratig series: Euler s trasformatio π , which equivaletly says 3. π , coverges very slowly. For example, the 00th partial sum of the series i 3. is , which is accurate to oly oe digit past the decimal poit. We will describe a method due to Euler for acceleratig the covergece of alteratig series, ad illustrate it for both 3. ad the alteratig harmoic series 3. l Euler s basic idea is that a coverget alteratig series 3.3 S a 0 a + a a 3 + a 4 a 5 + ca be rewritte as 3.4 S a 0 + a0 a a a a + a 3 a3 a 4 a4 + a 5, where each term of the origial series is split i half ad combied with half of the adjacet terms o both sides of the origial series except the first term a 0, where a sigle a 0 / is left o its ow. The order of additio has ot chaged i passig from 3.3 to 3.4, so the value of the series does ot chage. Sice the terms a /a + / a a + / may have a faster decay rate tha the origial terms a, applyig this trasformatio multiple times ca accelerate the covergece i a impressive way. Example 3.. Applyig to 3. turs this series ito π We have chaged 3. ito 3.5 π + by replacig a a a a + i 3.3 with Euler gives a brief accout of acceleratig the series i our Examples 3. ad 3. i [, Chap., Part II] pp of the origial Lati ad pp i the Eglish traslatio. See also [4, Sect. 35B].

8 8 KEITH CORAD ow view the alteratig series i 3.5 as a istace of 3.3 ad trasform it usig 3.4: 4 π , which has chaged 3.5 ito 3.7 π by replacig a i 3.6 with 3.8 a a a ext view the alteratig series i 3.7 as 3.3 ad trasform it usig 3.4: π We have chaged 3.7 ito 3.9 π by replacig a i 3.8 with a a a Applyig this process two more times, we get 3. π ad 3. π

9 ACCELERATIG COVERGECE OF SERIES 9 We ow have six partial sums that each ted to π as : s s + 4 +, s , , s , s , s The table below lists these partial sums at 0, 0, 5, 50, 00, ad 000 trucated ot rouded to 8 digits after the decimal poit. While s 0 is oly accurate to oe digit, s 5 0 is accurate to 6 digits. While s 00 is oly accurate to two digits, s 5 00 is accurate to digits the 9th ad 0th digits after the decimal poit are ot i the table s s s s s s The reaso accelerated series ted to coverge faster is that their terms decay to 0 at ever faster rates. Terms i the successive series for π 3., 3.5, 3.7, 3.9, 3., ad 3. decay as follows: a 4 +, a , a , a a 4 a , ,

10 0 KEITH CORAD I geeral, after applyig k series acceleratios to 3. we have 3.3 π k j0 j! 3 5 j + + 4k! k +, where the first sum i 3.3 is 0 for k 0. This formula at k 0,,, 3, 4, ad 5 is 3., 3.5, 3.7, 3.9, 3., ad 3. respectively, ad for each k the magitude of the th term i the secod series i 3.3 decays like / k+ up to a scalig factor: as, 4k! k + 4k! k!/k k+ k+. I a aswer o the series 3. is accelerated 4 times. Error bouds o the remaider for each series for π ca be obtaied from the alteratig series test: the absolute value of the first omitted term is a boud. Writig r i r < 4 + <, r < 4 4, r < , r < , r4 < , r5 < π si, For example, r 4 < if 3/ 5 <.00000, which is the same as 0. Thus π is betwee s ad s Example 3.. ow we tur to the alteratig harmoic series 3., ad will be more brief tha we were with the series for π. Write 3. as a a + a 3 a 4 + a, where a. Acceleratig 3. oce turs that series ito 3.4 a + a a Acceleratig 3.4 makes it ad the reader should check as a exercise that the ext few acceleratios of 3. are ad , , I the table below we list partial sums of 3. ad its accelerated forms 3.4, 3.5, 3.6, 3.7, ad 3.8. The otatio s i i the first colum is, by aalogy with the series for π, the ith accelerated form of 3., for 0 i 5, with the sum ruig up to.

11 ACCELERATIG COVERGECE OF SERIES s s s s s s Sice all these series fit the alteratig series test, we ca boud remaiders usig the first missig term. For example, from 3.8, r 5 0 5/ /0 8, so from the value of s 5 0 we kow 3. equals By compariso, the 000th partial sum of 3. is accurate to just two digits! Geeralizig the series , after k acceleratios 3.9 k j j j + k! k + + k, where the first sum o the right i 3.9 is 0 whe k 0, ad for each k the th term of the secod series o the right i 3.9 decays like / k+ up to a scalig factor: as, k! k + + k k!/k k+. We ca describe the effect of applyig Euler s trasformatio k times to a by usig otatio from the differece calculus. For a sequece a a 0, a, a,..., its first discrete differece is the sequece a a a 0, a a, a 3 a,..., so a a + a. The secod discrete differece of a is a a, which starts out as a a0, a a,... a a + a 0, a 3 a + a,... ad i geeral the kth discrete differece of a is k a k a. The formula a a + a + + a suggests a coectio with biomial coefficiets usig alteratig sigs. Ideed, for k 0 we have k k k a kj a +j j for each 0. I particular, k a0 j0 j0 k k kj a j a k ka k + + k a 0. j With this otatio, Euler s trasformatio i 3.4 cosists of rewritig 3.0 a 0 + a a + a 0 a + a a 0 a as a. Apply Euler s trasformatio to the series o the right i 3.0 gives us a a0 a a a 0 a,

12 KEITH CORAD ad feedig this ito 3.0 shows a 0 a a 0 a is a a 0 a a. I geeral, applyig Euler s trasformatio k times leads to the acceleratio formula 3. a k j0 j j+ j a0 + where the first fiite series o the right is 0 at k 0. k k a, Remark 3.3. While we are iterested i examples where Euler s trasformatio speeds up covergece, it does ot always have such a effect. For example, if a r with r < the a r + r r r r a, so Euler s trasformatio o a geometric series leads to o improvemet: r r r. A versio of Euler s trasformatio ca be applied to ay coverget series that ca be writte as a power series a c for some c i [,, ot just for alteratig series the case c : a c a 0 + a c + a c + a 3 c 3 + c a 0 c + a c c c + a c c 3 c + a c 3 c 4 3 c + a 0 c + a 0 + a c + a + a c c a 0 c + a a 0 c c + a a a 0 c + c c a 0 c + c c c + a + a 3 c 3 + c c c + a 3 a c c3 + a + a c ac. Whe c this is 3.0, ad i case the result seems like a trick it could also be derived usig summatio by parts with u a ad v c + /c. Repeatig this process k times for k 0, a c k j0 c j c j+ j a0 + c k c k k ac, where the first sum o the right is 0 at k 0. At c the above formula is 3.. For more o this, see [3] ad [4, Sect. 33, 35], but watch out: i [4], a a a +. That is the egative of our covetio, so i [4] is our.

13 ACCELERATIG COVERGECE OF SERIES 3 4. More speed-up methods We briefly metio two further techiques for acceleratig covergece of series. The Shaks trasformatio is applied to the partial sums of a series, ot to the terms of the series. If S a has partial sums s for the the Shaks trasformatio of the partial sums is the ew sequece s where s s + s + s + s + s + + s provided the deomiators are ot 0. For example, if S l / the s is oly accurate to oe digit while s is accurate to 3 digits ad s is accurate to 5 digits. The discrete Fourier trasform of a sequece a 0, a,..., a m i C is the ew sequece â 0, â,..., â m where â k m j0 a ke πijk/m. Calculatig all of these fiite series rapidly is a importat task. To compute each â k from its defiitio requires m multiplicatios, so computig every â k requires m m m multiplicatios. The fast Fourier trasform FFT is a alterate approach to computig the discrete Fourier trasform that requires somethig o the order of at most m l m operatios, which is a big improvemet o the aive approach directly from the defiitios. Refereces [] L. Euler, De summatioe iumerabilium progressioum, Comm. Acad. Sci. Petropol , Olie Eglish traslatio at [] L. Euler, Istitutioes calculi differetialis cum eius usu i aalysi fiitorum ac doctria serierum 755 Olie i Lati at ad i Eglish at 7ceturymaths.com/cotets/differetialcalculus.htm [3] J. Gal-Ezer ad G. Zwas, Covergece acceleratio as a computatioal assigmet, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology 8 987, 5 8. Olie at [4] K. Kopp, Theory ad Applicatio of Ifiite Series, Blackie ad So Ltd., Lodo, 95. Olie at [5] E. Kummer, Eie eue Methode, die umerische Summe lagsam covergireder Reihe zu bereche, J. Reie Agew. Math pp Olie at jouralfrdierei3crelgoog. [6] E. Sadifer, Estimatig the Basel Problem. Olie at HEDI-003-.pdf.