The acceleration formula for Zeta(, 5, ) is Theorem. Let the column-vector of functions of, x T a(x):=, [a[0](x), a[1](x), a[2](x), a[3](x)] , be defined T by the initial conditions:, a(0) = [1, 0, 0, 0] and a(x+1)=A(x)a(x), where A(x) is the, 4, by , 4, matrix 2 3 (3 x + 5) (12 x + 39 x + 32) (x + 1) [[--------------------------------------, 5 32 (5 x + 4) (2 x + 3) 5 4 3 2 3 (310 x + 1914 x + 4557 x + 5180 x + 2785 x + 568) (x + 1) - --------------------------------------------------------------, 5 32 (5 x + 4) (5 x + 3) (2 x + 3) 6 5 4 3 2 3 (465 x + 2995 x + 7638 x + 9771 x + 6556 x + 2186 x + 283) (x + 1) ------------------------------------------------------------------------, - 5 32 (5 x + 3) (5 x + 2) (2 x + 3) 8 7 6 5 4 3 (73625 x + 525650 x + 1553145 x + 2464536 x + 2285205 x + 1263475 x 2 4 / + 405060 x + 68390 x + 4616) (x + 1) / (160 (5 x + 1) (5 x + 2) / 2 3 5 3 (23 x + 76 x + 63) (x + 1) (5 x + 3) (5 x + 4) (2 x + 3) )], [------------------------------, 5 32 (5 x + 4) (2 x + 3) 4 3 2 3 (570 x + 2600 x + 4172 x + 2749 x + 637) (x + 1) - ----------------------------------------------------, 5 32 (5 x + 4) (5 x + 3) (2 x + 3) 5 4 3 2 3 (815 x + 3935 x + 7015 x + 5708 x + 2141 x + 298) (x + 1) 7 --------------------------------------------------------------, - (24000 x 5 32 (5 x + 3) (5 x + 2) (2 x + 3) 6 5 4 3 2 + 132625 x + 291175 x + 327920 x + 204853 x + 70907 x + 12532 x + 868 4 / 5 ) (x + 1) / (32 (5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4) (2 x + 3) )], [ / 3 3 2 3 15 (3 x + 5) (x + 1) (355 x + 1039 x + 893 x + 239) (x + 1) -----------------------, - -----------------------------------------, 5 5 32 (5 x + 4) (2 x + 3) 32 (5 x + 4) (5 x + 3) (2 x + 3) 4 3 2 3 (480 x + 1535 x + 1614 x + 693 x + 104) (x + 1) ---------------------------------------------------, - 5 32 (5 x + 3) (5 x + 2) (2 x + 3) 6 5 4 3 2 (12875 x + 50225 x + 73870 x + 53354 x + 20020 x + 3686 x + 260) 4 / 5 (x + 1) / (32 (5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4) (2 x + 3) )], [ / 3 3 5 (x + 1) 5 (3 x + 2) (x + 1) -----------------------, - -----------------------, 5 5 16 (5 x + 4) (2 x + 3) 32 (5 x + 4) (2 x + 3) 2 3 2 4 (19 x + 18 x + 4) (x + 1) (3 x + 2) (15 x + 12 x + 2) (x + 1) ---------------------------, - -------------------------------------]] 5 5 32 (5 x + 2) (2 x + 3) 16 (5 x + 2) (5 x + 4) (2 x + 3) T In other words, a(x)=A(x-1)A(x-2)...A(0) times, [1, 0, 0, 0] Let 3 2 a[0](x) (41 x + 33) a[1](x) (5 x + 22 x + 27 x + 10) b(x) = 1/8 ------------------- + 1/16 ---------------------------------- 5 x + 4 (5 x + 4) (5 x + 3) 4 3 2 a[2](x) (15 x + 30 x - 2 x - 30 x - 13) 7 - 1/32 ------------------------------------------ + 1/160 a[3](x) (2375 x (5 x + 3) (5 x + 2) 6 5 4 3 2 + 8775 x + 14270 x + 16586 x + 17319 x + 13311 x + 5684 x + 976)/( (5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4)) We have the following acceleration formula (with convergence-rate, 0.0004349673298, ) infinity infinity ----- ----- \ 1 \ ) -------- = ) b(x) / 5 / ----- (z + 1) ----- z = 0 x = 0 Proof: The following is a Markov-WZ pair 5 2 3 ((z + x)!) (3 x + 5) (12 x + 39 x + 32) (x + 1) [-----------------, [[--------------------------------------, 5 5 x + 4 ((2 x + z + 1)!) 5 4 3 2 3 (310 x + 1914 x + 4557 x + 5180 x + 2785 x + 568) (x + 1) - --------------------------------------------------------------, (5 x + 4) (5 x + 3) 6 5 4 3 2 3 (465 x + 2995 x + 7638 x + 9771 x + 6556 x + 2186 x + 283) (x + 1) ------------------------------------------------------------------------, - (5 x + 3) (5 x + 2) 8 7 6 5 4 3 (73625 x + 525650 x + 1553145 x + 2464536 x + 2285205 x + 1263475 x 2 4 + 405060 x + 68390 x + 4616) (x + 1) /(5 (5 x + 1) (5 x + 2) (5 x + 3) 2 3 3 (23 x + 76 x + 63) (x + 1) (5 x + 4))], [------------------------------, 5 x + 4 4 3 2 3 (570 x + 2600 x + 4172 x + 2749 x + 637) (x + 1) - ----------------------------------------------------, (5 x + 4) (5 x + 3) 5 4 3 2 3 (815 x + 3935 x + 7015 x + 5708 x + 2141 x + 298) (x + 1) 7 --------------------------------------------------------------, - (24000 x (5 x + 3) (5 x + 2) 6 5 4 3 2 + 132625 x + 291175 x + 327920 x + 204853 x + 70907 x + 12532 x + 868 4 ) (x + 1) /((5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4))], [ 3 3 2 3 15 (3 x + 5) (x + 1) (355 x + 1039 x + 893 x + 239) (x + 1) ---------------------, - -----------------------------------------, 5 x + 4 (5 x + 4) (5 x + 3) 4 3 2 3 (480 x + 1535 x + 1614 x + 693 x + 104) (x + 1) ---------------------------------------------------, - (5 x + 3) (5 x + 2) 6 5 4 3 2 (12875 x + 50225 x + 73870 x + 53354 x + 20020 x + 3686 x + 260) 3 4 10 (x + 1) (x + 1) /((5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4))], [-----------, 5 x + 4 3 2 3 5 (3 x + 2) (x + 1) (19 x + 18 x + 4) (x + 1) - --------------------, ---------------------------, 5 x + 4 5 x + 2 2 4 2 (3 x + 2) (15 x + 12 x + 2) (x + 1) 2 3 - ---------------------------------------]], [1, z, z , z ], [(3960 (5 x + 2) (5 x + 4) 2 3 4 5 + 10230 z + 61020 x + 389800 x + 1377800 x + 3003000 x + 4218560 x 2 2 3 2 + 147325 x z + 10950 z + 146575 x z + 2694325 x z + 858525 x z 2 2 3 4 3 2 2 3 + 770150 x z + 77850 x z + 5043625 x z + 2102375 x z + 363050 x z 4 3 4 5 7 6 + 23750 x z + 6300 z + 2100 z + 390 z + 2196600 x + 3841720 x 8 9 7 6 2 5 3 + 718000 x + 102500 x + 1581375 x z + 1396250 x z + 631250 x z 3 6 6 4 4 3 5 8 + 625 x z + 30 z + 153750 x z + 18625 x z + 263125 x z 7 2 6 3 5 4 4 5 6 + 278125 x z + 156250 x z + 50000 x z + 8750 x z + 275 x z 2 6 5 4 2 3 3 2 4 + 750 x z + 5818255 z x + 3265850 x z + 836500 x z + 96150 x z 5 6 5 2 4 3 3 4 + 3995 x z + 4065775 x z + 2916125 x z + 1020000 x z + 178250 x z 2 5 / + 13600 x z ) / (5 (5 x + 1) (5 x + 2) (5 x + 3) (5 x + 4) / 5 10 2 3 (2 x + z + 2) ), (200 + 5030 z + 3040 x + 1250 x + 19190 x + 67500 x 4 5 2 2 + 148300 x + 214300 x + 76175 x z + 12590 z + 178275 x z 3 2 2 2 3 4 + 1648675 x z + 476750 x z + 1017800 x z + 179700 x z + 3514900 x z 3 2 2 3 4 3 4 5 + 3124250 x z + 920275 x z + 97550 x z + 13700 z + 8100 z + 2750 z 7 6 8 9 7 + 132740 x + 207240 x + 53900 x + 12500 x + 2416325 x z 6 2 5 3 3 6 2 7 4 6 + 4420350 x z + 3202750 x z + 20250 x z + 875 x z + 8750 x z 3 7 6 7 4 4 3 5 8 + 625 x z + 510 z + 40 z + 1141250 x z + 204025 x z + 775250 x z 7 2 6 3 5 4 4 5 + 1686500 x z + 1494375 x z + 680000 x z + 166250 x z 9 8 2 7 3 6 4 5 5 + 108750 x z + 275625 x z + 290625 x z + 162500 x z + 51250 x z 6 2 6 7 5 4 2 + 5025 x z + 16075 x z + 350 x z + 4832155 z x + 5719475 x z 3 3 2 4 5 6 5 2 + 2442125 x z + 439550 x z + 30115 x z + 4310150 x z + 6456815 x z 4 3 3 4 2 5 / + 3686850 x z + 973850 x z + 116390 x z ) / (5 (5 x + 1) (5 x + 2) / 5 10 (5 x + 3) (5 x + 4) (2 x + z + 2) ), (260 + 1880 z + 3525 x - 15000 x 2 3 4 5 2 + 19140 x + 55325 x + 90900 x + 74910 x + 24450 x z + 9680 z 2 3 11 2 2 2 + 131600 x z + 347125 x z - 1875 x + 126125 x z + 728675 x z 3 4 3 2 2 3 4 + 257700 x z + 557900 x z + 2235325 x z + 1341550 x z + 248100 x z 3 4 5 7 6 8 + 19760 z + 20400 z + 11960 z - 81210 x - 4700 x - 90600 x 7 4 8 3 9 7 6 2 + 8750 z x + 625 z x - 50675 x - 61925 x z + 4328675 x z 5 3 3 6 2 7 8 4 6 + 6814810 x z + 230075 x z + 19275 x z + 475 x z + 175000 x z 3 7 2 8 6 7 8 4 4 + 21875 x z + 1000 x z + 4080 z + 760 z + 60 z + 4045050 x z 3 5 8 7 2 6 3 + 1121875 x z - 113800 x z + 2248025 x z + 4432100 x z 5 4 4 5 9 8 2 + 3333750 x z + 1232200 x z - 50625 x z + 676250 x z 7 3 6 4 5 5 6 2 6 + 1622875 x z + 1492500 x z + 698125 x z + 41200 x z + 143425 x z 7 5 4 2 3 3 2 4 + 6850 x z + 514255 z x + 4264000 x z + 3780775 x z + 1165050 x z 5 6 5 2 4 3 + 133450 x z + 209300 x z + 5302330 x z + 6425875 x z 3 4 2 5 6 5 3 8 4 7 + 2859900 x z + 548665 x z + 51250 z x + 256875 z x + 281250 z x 10 2 9 5 6 / - 8125 z x + 90000 z x + 160625 z x ) / (5 (5 x + 1) (5 x + 2) / 5 10 (5 x + 3) (5 x + 4) (2 x + z + 2) ), (976 + 7180 z + 10564 x + 81895 x 2 3 4 5 2 + 51491 x + 150474 x + 298011 x + 432896 x + 72540 x z + 23380 z 2 3 11 2 2 2 + 220670 x z + 859105 x z + 20650 x + 325565 x z + 912360 x z 3 4 3 2 2 3 4 + 449260 x z + 1492950 x z + 2174200 x z + 1863900 x z + 579100 x z 3 4 5 7 6 8 + 48040 z + 63000 z + 52276 z + 454716 x + 491414 x + 342574 x 12 7 4 7 5 8 4 8 3 + 2375 x + 183750 z x + 51250 z x + 8750 z x + 23500 z x 9 9 3 7 6 2 5 3 + 199436 x + 625 z x + 1183215 x z + 2411945 x z + 7744790 x z 3 6 2 7 8 4 6 3 7 + 1338700 x z + 182950 x z + 9950 x z + 1349750 x z + 260700 x z 2 8 9 6 7 8 9 + 23200 x z + 650 x z + 27300 z + 8720 z + 1560 z + 120 z 4 4 3 5 8 7 2 + 8213800 x z + 3661094 x z + 667330 x z + 1226710 x z 6 3 5 4 4 5 9 + 5733860 x z + 7940650 x z + 4676186 x z + 282725 x z 8 2 7 3 6 4 5 5 + 455400 x z + 2821650 x z + 4863250 x z + 3599645 x z 6 2 6 7 5 4 2 + 220950 x z + 745100 x z + 63500 x z + 1827455 z x + 3311450 x z 3 3 2 4 5 6 + 4532950 x z + 2350200 x z + 457734 x z + 1664390 x z 5 2 4 3 3 4 2 5 + 3391455 x z + 7179100 x z + 5518000 x z + 1730486 x z 9 2 11 6 5 6 6 3 8 + 1125 z x + 10000 z x + 723125 z x + 160625 z x + 837500 z x 3 9 4 7 4 8 10 2 10 + 113750 z x + 1721250 z x + 268750 z x + 77625 z x + 15625 z x 2 9 5 7 5 6 / + 117125 z x + 283625 z x + 1543150 z x ) / (5 (5 x + 1) (5 x + 2) / 5 (5 x + 3) (5 x + 4) (2 x + z + 2) )]] (check!) Now use formula (Acc) in the human-paper The Markov-WZ Method, QED The error in using 30 terms is 0.13266857703099894443058632192714554688915305699356137063613466179426018456\ -103 3518463157978365299163 10