ACCELERATING CONVERGENCE OF SERIES

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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.64. 0 0 5 50 00 000 s.54976773.596634.6057340.65373.63498390.64393456 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 0000. 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 <.000 + > 0000, so 0000.

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.3 + + + + +, 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. 0 0 5 50 00 000 s.6406768.643789.6448494.64474057.64488489.64493356 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 7.644837..., the series. lies betwee s 7.000.644737... ad s 7 +.00.644937..., ad sice /.0000005 whe 000, the value of s 000 tells us. is.64493 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 / + / + + 4 / + + 4.

I. with a ACCELERATIG COVERGECE OF SERIES 3 + ad b + 4 + + Feedig this ito the right side of.3,.6 Whe s + 4 + previous tables for s ad s + 4 + + +, we have B 4 ad b a + 4 + + +. + +. + :, the ext table exhibits faster covergece tha + + for the same values of. 0 0 5 50 00 000 s.64446470.64486454.6448979.64499.6449334.64493406 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 0.000.64476... ad s 0 +.000.644964...: 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 + + + 3, where /3.7 B b + + /3 + + + 3 9 ad b a, so. tells us + 3 + + Feedig this ito.6,.8 9 + 9 + + 4 + 9 + + 3 + + 6 + + + 3. 6 + + + 3. Settig s + 4 + 9 + 6, we have the followig values. + + + 3

4 KEITH CORAD 0 0 5 50 00 000.6448530.644978.644930.64493385.64493405.64493406 s We have s + r where r r < + + 6 + + + 3 6 5 < 6 x 5 dx 6 4 4 3 4, has the boud so r 5 <.00000384. Usig the table above,. is betwee s 5.00000384 >.644976 ad s 5 +.00000384 <.64493495, 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+. + 0 + + + 3 + 4 + 5. 0 0 5 50 00 000 s 5.6449895.6449339.6449340.64493406.64493406.64493406 By., / s 5 0 r5 0 < 0/6/06 /0 5.00000035, which puts. betwee s 5 0.00000035.6449335... ad s5 0 +.00000035.644934.... The series. that we have bee fidig good approximatios to has a exact formula: π.6449340.... 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.644934.

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. 4 3 4 + b a a 4 + 3 + 3 + +. ow let a 3 + 3 + +, so a grows like rate whose exact sum is kow is b b so by. ad algebra 3 4 + 6 + 3. A sequece growig at the same 4 3 : by.9, + + + 3 3 + + + 3 3 3 3! 6, b a a 4 + 6 + + 6 3 + + + 3. + 6 ext let a 3, which grows like. A sequece growig at the + + + 3 5 same rate whose exact sum is kow is b : by.9, + + + 3 + 4 b so by. ad algebra + + + 3 + 4 4 4! 96, 3 4 + 6 + 96 + b a a 4 + 6 + 96 + It is left to the reader to derive the ext acceleratio, which is 3 4 + 6 + 96 + + 74 + 0 3 + + 5. We ow have five partial sums that each ted to s 3, s 4 + 3 + 3 + +, as : 3 s 4 + 6 + s 4 + 6 + 96 + 50 + 4 3 + + + 3 + 4, 50 + 4 3 + + 4. + 6 3 + + + 3, s 4 4 + 6 + 96 + + 74 + 0 3 + + + 3 + 4 + 5.

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. 0 0 5 50 00 000 s.975398.0086784.0886.086086.000740.005640 s.03986.095009.00005.004940.005593.005690 s.0906.00440.00538.00565.005687.005690 s.00708.005498.0056.005687.005690.005690 s 4.004483.005655.005679.005690.005690.005690 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 : + + + 3 + 3 + + < + 6 3 + + + 3 < + + 50 + 4 3 + + + 3 + 4 < 74 + 0 3 + + + 5 < 3 3 + < 3 + + 3 < + + 50 6 < 74 7 < 3 x 4 dx 3, dx x5 4 4, 50 50 dx x6 5 5 0 5, 74 74 dx x7 6 6 37 3 6. 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 3 0.0000.0044... ad s 0 +.0000.0064.... We also have r 4 <.00000 for 9, so lies betwee s4 3 0.00000.00555... ad s 4 0 +.00000.00575.... 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 +! 3 + + 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.

The Leibiz series ACCELERATIG COVERGECE OF SERIES 7 3. Alteratig series: Euler s trasformatio π 4 3 + 5 7 + 9 + 3 5 + 7 9 + +, which equivaletly says 3. π 4 4 3 + 4 5 4 7 + 4 9 4 + 4 3 4 5 + 4 7 4 9 + 4 +, coverges very slowly. For example, the 00th partial sum of the series i 3. is 3.5..., 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 + 3 4 + 5 6 + 7 8 + 9 0 +. 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 3.3 3.4 to 3. turs this series ito π + 3 3 + 5 5 7 7 + 9 9 + 4 3 4 3 5 + 4 5 7 4 7 9 + 4 9 4 3 +. We have chaged 3. ito 3.5 π + by replacig a 4 + 3.6 a a a + i 3.3 with 4 + + 3 + + 3 + + 3 + + 3 + 3 4 + + 3. Euler gives a brief accout of acceleratig the series i our Examples 3. ad 3. i [, Chap., Part II] pp. 36-37 of the origial Lati ad pp. 399-400 i the Eglish traslatio. See also [4, Sect. 35B].

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 + 3.4 + 3 + 3 4 3 5 + 4 5 7 4 7 9 + 4 9 4 3 3 5 3 5 + 5 7 3 + 5 7 7 9 + 3 + 8 3 5 8 3 5 7 + 8 5 7 9 8 7 9 +, which has chaged 3.5 ito 3.7 π + 3 + by replacig a i 3.6 with 3.8 a a a + 8 + + 3 + 5 7 9 + 9 4 + + 3 4 + 3 + 5 8 + + 3 + 5. ext view the alteratig series i 3.7 as 3.3 ad trasform it usig 3.4: π + 3 + 8 3 5 8 3 5 7 + 8 5 7 9 8 3.4 + 3 + 4 5 + 4 3 5 4 3 5 7 + 3 + 4 5 + 4 3 5 7 4 3 5 7 9 +. We have chaged 3.7 ito 3.9 π + 3 + 4 5 + by replacig a i 3.8 with 7 9 + 4 3 5 7 4 5 7 4 + + 3 + 5 + 7 + + 3.0 a a a 4 + + 3 + 5 + 7. Applyig this process two more times, we get 3. π + 3 + 4 5 + 05 + 96 + + 3 + 5 + 7 + 9 ad 3. π + 3 + 4 5 + 05 + 48 945 + 480 + + 3 + 5 + 7 + 9 +.

ACCELERATIG COVERGECE OF SERIES 9 We ow have six partial sums that each ted to π as : s s + 4 +, s + 3 + 4 + + 3, 8 + + 3 + 5, s + 3 + 4 5 + 4 + + 3 + 5 + 7, s 4 + 3 + 4 5 + 05 + 96 + + 3 + 5 + 7 + 9, s 5 + 3 + 4 5 + 05 + 48 945 + 480 + + 3 + 5 + 7 + 9 +. 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. 0 0 5 50 00 000 s 3.33580 3.898478 3.03453 3.6986 3.549340 3.45965 s 3.453598 3.46735 3.40886 3.4783 3.4648 3.45935 s 3.4880 3.463956 3.45676 3.45960 3.4593 3.45965 s 3.46337 3.459558 3.45934 3.45975 3.45966 3.45965 s 4 3.45967 3.45988 3.45956 3.45965 3.45965 3.45965 s 5 3.45938 3.45967 3.45964 3.45965 3.45965 3.45965 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 4 + + 3, a 8 + + 3 + 5 3, a a 4 a 5 4 + + 3 + 5 + 7 3 4, 96 + + 3 + 5 + 7 + 9 3 5, 480 + + 3 + 5 + 7 + 9 + 5 6.

0 KEITH CORAD I geeral, after applyig k series acceleratios to 3. we have 3.3 π k j0 j! 3 5 j + + 4k! + + 3 + 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! + + 3 + k + 4k! k!/k k+ k+. I a aswer o https://math.stackexchage.com/questios/70694/why-is-theleibiz-method-for-approximatig-pi-so-iefficiet 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 < 8 8 3 3, r < 4 6 4 3 4, r4 < 96 3 5 3 5, r5 < 480 64 6 5 6. π si, For example, r 4 <.00000 if 3/ 5 <.00000, which is the same as 0. Thus π is betwee s 4 0.00000 3.459... ad s4 0 +.00000 3.4593.... 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 + + + + 4 + 4 40 +. Acceleratig 3.4 makes it 3.5 + 8 + 4 + 4 + + + 8 + + + ad the reader should check as a exercise that the ext few acceleratios of 3. are 3.6 3.7 ad 3.8 + 8 + 4 + 3 4 + + + 3, + 8 + 4 + 64 + 3 + + + 3 + 4, + 8 + 4 + 64 + 60 + 5 4 + + + 3 + 4 + 5. 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.

ACCELERATIG COVERGECE OF SERIES 0 0 5 50 00 000 s.6456349.6687740.774749.683476.68877.6964743 s.6908946.695809.6935673.6930508.69367.6934693 s.6998340.69305.6936060.6934535.6934694.693478 s.693909.6934557.6934788.69347.693477.693478 s 4.6934470.6934705.69347.693477.693478.693478 s 5.6934678.693476.693478.693478.693478.693478 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 5/46 4.37/0 8, so from the value of s 5 0 we kow 3. equals.69347.... By compariso, the 000th partial sum of 3. is accurate to just two digits! Geeralizig the series 3.4 3.8, 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,

KEITH CORAD ad feedig this ito 3.0 shows a 0 a a 0 a is a a 0 a + 4 4 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.

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.693478... / the s 00.6887... is oly accurate to oe digit while s 0.693065... is accurate to 3 digits ad s 50.69346... 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. 5 738, 9 05. Olie Eglish traslatio at http://www.7ceturymaths.com/cotets/euler/e00tr.pdf. [] L. Euler, Istitutioes calculi differetialis cum eius usu i aalysi fiitorum ac doctria serierum 755 Olie i Lati at http://eulerarchive.maa.org/pages/e.html ad i Eglish at http://www. 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 https:// www.tadfolie.com/doi/pdf/0.080/0007398708003. [4] K. Kopp, Theory ad Applicatio of Ifiite Series, Blackie ad So Ltd., Lodo, 95. Olie at https://archive.org/details/theoryadapplica0369mbp. [5] E. Kummer, Eie eue Methode, die umerische Summe lagsam covergireder Reihe zu bereche, J. Reie Agew. Math. 6 837 pp. 06-4. Olie at https://archive.org/details/ jouralfrdierei3crelgoog. [6] E. Sadifer, Estimatig the Basel Problem. Olie at http://eulerarchive.maa.org/hedi/ HEDI-003-.pdf.

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