First Version: May 28, 2004: tested for Maple 8 and 9 Version of Aug. 2, 2004: Adding Verbose Modes of Markov and ACC, called MarkovPaper, ACCpaper This is MarkovWZ, A Maple package accompanying the article: The WZ-Markov Method by M. Mohammed and D. Zeilberger The most current version is available on WWW at: http://www.math.rutgers.edu/~zeilberg . Please report all bugs to: zeilberg at math dot rutgers dot edu . For general help, and a list of the available functions, type "ezra();". For specific help type "ezra(procedure_name)" The acceleration formula for Zeta(, 6, ) is Warning, the protected names norm and trace have been redefined and unprotected Theorem. Let the column-vector of functions of, x T a(x):=, [a[0](x), a[1](x), a[2](x), a[3](x), a[4](x)] , be defined T by the initial conditions:, a(0) = [1, 0, 0, 0, 0] and a(x+1)=A(x)a(x), where A(x) is the, 5, by , 5, matrix 4 3 2 3 (201 x + 1308 x + 3201 x + 3492 x + 1433) (x + 1) [[-----------------------------------------------------, - 6 128 (6 x + 5) (2 x + 3) 6 5 4 3 2 (6615 x + 51786 x + 165405 x + 274632 x + 248659 x + 115802 x + 21646) 3 / 6 7 6 (x + 1) / (768 (2 + 3 x) (6 x + 5) (2 x + 3) ), (6615 x + 54880 x / 5 4 3 2 + 189897 x + 353622 x + 380795 x + 235940 x + 77578 x + 10408) 3 / 6 9 (x + 1) / (768 (6 x + 5) (1 + 2 x) (2 x + 3) ), - (136269 x / 8 7 6 5 4 + 1214325 x + 4640937 x + 9947733 x + 13134545 x + 11047425 x 3 2 4 / + 5907244 x + 1933390 x + 351000 x + 26912) (x + 1) / (1536 (1 + 2 x) / 6 11 10 (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + 3) ), (154791 x + 1514436 x 9 8 7 6 5 + 6476645 x + 15879528 x + 24584922 x + 24925238 x + 16576183 x 4 3 2 3 / + 7016252 x + 1747215 x + 208168 x + 2786 x - 940) (x + 1) / (1536 / 6 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1) (2 x + 3) )], [ 2 3 7 (3 x + 5) (12 x + 39 x + 32) (x + 1) ----------------------------------------, 6 64 (6 x + 5) (2 x + 3) 5 4 3 2 3 (2652 x + 16535 x + 39893 x + 46184 x + 25464 x + 5360) (x + 1) - --------------------------------------------------------------------, 7 6 128 (2 + 3 x) (6 x + 5) (2 x + 3) 6 5 4 3 2 (1080 x + 7233 x + 19401 x + 26498 x + 19320 x + 7110 x + 1030) 3 / 6 8 7 (x + 1) / (384 (6 x + 5) (1 + 2 x) (2 x + 3) ), - (47208 x + 344071 x / 6 5 4 3 2 + 1046358 x + 1727255 x + 1688192 x + 999194 x + 349424 x + 65942 x 4 / + 5140) (x + 1) / (256 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5) / 6 10 9 8 7 6 (2 x + 3) ), (92232 x + 702135 x + 2226819 x + 3738750 x + 3374986 x 5 4 3 2 + 1201443 x - 544659 x - 758126 x - 321624 x - 60918 x - 4190) 3 / 6 (x + 1) / (768 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1) (2 x + 3) )], [ / 2 3 21 (23 x + 76 x + 63) (x + 1) -------------------------------, 6 128 (6 x + 5) (2 x + 3) 4 3 2 3 7 (693 x + 3204 x + 5245 x + 3560 x + 856) (x + 1) - ------------------------------------------------------, 6 256 (2 + 3 x) (6 x + 5) (2 x + 3) 5 4 3 2 3 (4339 x + 22058 x + 42203 x + 37714 x + 15776 x + 2476) (x + 1) --------------------------------------------------------------------, - ( 6 256 (6 x + 5) (1 + 2 x) (2 x + 3) 7 6 5 4 3 2 71505 x + 403767 x + 915905 x + 1079691 x + 715048 x + 265912 x 4 / + 51276 x + 3944) (x + 1) / (512 (1 + 2 x) (1 + 3 x) (2 + 3 x) / 6 9 8 7 6 (6 x + 5) (2 x + 3) ), - (4557 x + 93394 x + 491965 x + 1251216 x 5 4 3 2 + 1827597 x + 1621300 x + 875225 x + 275586 x + 45332 x + 2900) 3 / 6 (x + 1) / (512 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1) (2 x + 3) )], [ / 3 3 2 3 35 (3 x + 5) (x + 1) 7 (429 x + 1283 x + 1144 x + 320) (x + 1) -----------------------, - --------------------------------------------, 6 6 64 (6 x + 5) (2 x + 3) 384 (2 + 3 x) (6 x + 5) (2 x + 3) 4 3 2 3 7 (357 x + 1233 x + 1450 x + 704 x + 120) (x + 1) -----------------------------------------------------, - 6 384 (6 x + 5) (1 + 2 x) (2 x + 3) 6 5 4 3 2 (34257 x + 136390 x + 207165 x + 154976 x + 59936 x + 11172 x + 760) 4 / 6 (x + 1) / (768 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + 3) ), - ( / 8 7 6 5 4 3 42525 x + 267813 x + 708558 x + 1038742 x + 915983 x + 490757 x 2 3 / + 153466 x + 25100 x + 1600) (x + 1) / (768 (6 x + 5) (1 + 2 x) / 3 6 35 (x + 1) (1 + 3 x) (6 x + 1) (2 x + 3) )], [------------------------, 6 128 (6 x + 5) (2 x + 3) 3 3 2 3 35 (2 + 3 x) (x + 1) 7 (103 x + 186 x + 108 x + 20) (x + 1) - ------------------------, -----------------------------------------, 6 6 256 (6 x + 5) (2 x + 3) 768 (6 x + 5) (1 + 2 x) (2 x + 3) 4 7 (2 + 3 x) (13 x + 8) (x + 1) x - ----------------------------------, - 6 512 (6 x + 5) (1 + 2 x) (2 x + 3) 6 5 4 3 2 (8821 x + 33939 x + 55191 x + 47329 x + 21906 x + 4980 x + 400) 3 / 6 (x + 1) / (1536 (6 x + 1) (1 + 2 x) (6 x + 5) (2 x + 3) )]] / T In other words, a(x)=A(x-1)A(x-2)...A(0) times, [1, 0, 0, 0, 0] Let a[0](x) (777 x + 649) b(x) = 1/128 --------------------- 6 x + 5 3 2 a[1](x) (105 x + 419 x + 496 x + 182) + 1/768 --------------------------------------- (6 x + 5) (2 + 3 x) 4 3 2 a[2](x) (105 x + 245 x + 82 x - 142 x - 84) - 1/768 ---------------------------------------------- + 1/1536 a[3](x) ( (6 x + 5) (1 + 2 x) 7 6 5 4 3 2 2163 x + 8533 x + 14833 x + 17141 x + 16156 x + 11318 x + 4624 x 8 + 784)/((1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5)) - 1/1536 a[4](x) (2457 x 7 6 5 4 3 2 + 9345 x + 15778 x + 14168 x + 895 x - 15111 x - 17758 x - 8570 x - 1540)/((6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1)) We have the following acceleration formula (with convergence-rate, 0.0002441406250, ) infinity infinity ----- ----- \ 1 \ ) -------- = ) b(x) / 6 / ----- (1 + z) ----- z = 0 x = 0 Proof: The following is a Markov-WZ pair 6 4 3 2 3 ((z + x)!) (201 x + 1308 x + 3201 x + 3492 x + 1433) (x + 1) [-----------------, [[-----------------------------------------------------, - 6 2 (6 x + 5) ((2 x + z + 1)!) 6 5 4 3 2 (6615 x + 51786 x + 165405 x + 274632 x + 248659 x + 115802 x + 21646) 3 7 6 5 (x + 1) /(12 (2 + 3 x) (6 x + 5)), (6615 x + 54880 x + 189897 x 4 3 2 3 + 353622 x + 380795 x + 235940 x + 77578 x + 10408) (x + 1) /(12 9 8 7 6 (6 x + 5) (1 + 2 x)), - (136269 x + 1214325 x + 4640937 x + 9947733 x 5 4 3 2 + 13134545 x + 11047425 x + 5907244 x + 1933390 x + 351000 x + 26912) 4 11 (x + 1) /(24 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5)), (154791 x 10 9 8 7 6 + 1514436 x + 6476645 x + 15879528 x + 24584922 x + 24925238 x 5 4 3 2 + 16576183 x + 7016252 x + 1747215 x + 208168 x + 2786 x - 940) 3 (x + 1) /(24 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1))], [ 2 3 7 (3 x + 5) (12 x + 39 x + 32) (x + 1) ----------------------------------------, 6 x + 5 5 4 3 2 3 (2652 x + 16535 x + 39893 x + 46184 x + 25464 x + 5360) (x + 1) - --------------------------------------------------------------------, 7 2 (2 + 3 x) (6 x + 5) 6 5 4 3 2 (1080 x + 7233 x + 19401 x + 26498 x + 19320 x + 7110 x + 1030) 3 8 7 6 (x + 1) /(6 (6 x + 5) (1 + 2 x)), - (47208 x + 344071 x + 1046358 x 5 4 3 2 + 1727255 x + 1688192 x + 999194 x + 349424 x + 65942 x + 5140) 4 10 (x + 1) /(4 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5)), (92232 x 9 8 7 6 5 + 702135 x + 2226819 x + 3738750 x + 3374986 x + 1201443 x 4 3 2 3 - 544659 x - 758126 x - 321624 x - 60918 x - 4190) (x + 1) /(12 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1))], [ 2 3 21 (23 x + 76 x + 63) (x + 1) -------------------------------, 2 (6 x + 5) 4 3 2 3 7 (693 x + 3204 x + 5245 x + 3560 x + 856) (x + 1) - ------------------------------------------------------, 4 (2 + 3 x) (6 x + 5) 5 4 3 2 3 (4339 x + 22058 x + 42203 x + 37714 x + 15776 x + 2476) (x + 1) --------------------------------------------------------------------, - ( 4 (6 x + 5) (1 + 2 x) 7 6 5 4 3 2 71505 x + 403767 x + 915905 x + 1079691 x + 715048 x + 265912 x 4 + 51276 x + 3944) (x + 1) /(8 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5)), - 9 8 7 6 5 4 (4557 x + 93394 x + 491965 x + 1251216 x + 1827597 x + 1621300 x 3 2 3 + 875225 x + 275586 x + 45332 x + 2900) (x + 1) /(8 (6 x + 5) (1 + 2 x) 3 35 (3 x + 5) (x + 1) (1 + 3 x) (6 x + 1))], [---------------------, 6 x + 5 3 2 3 7 (429 x + 1283 x + 1144 x + 320) (x + 1) - --------------------------------------------, 6 (2 + 3 x) (6 x + 5) 4 3 2 3 7 (357 x + 1233 x + 1450 x + 704 x + 120) (x + 1) -----------------------------------------------------, - 6 (6 x + 5) (1 + 2 x) 6 5 4 3 2 (34257 x + 136390 x + 207165 x + 154976 x + 59936 x + 11172 x + 760) 4 8 (x + 1) /(12 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5)), - (42525 x 7 6 5 4 3 2 + 267813 x + 708558 x + 1038742 x + 915983 x + 490757 x + 153466 x 3 + 25100 x + 1600) (x + 1) /(12 (6 x + 5) (1 + 2 x) (1 + 3 x) (6 x + 1))], 3 3 35 (x + 1) 35 (2 + 3 x) (x + 1) [-----------, - ---------------------, 2 (6 x + 5) 4 (6 x + 5) 3 2 3 7 (103 x + 186 x + 108 x + 20) (x + 1) -----------------------------------------, 12 (6 x + 5) (1 + 2 x) 4 7 (2 + 3 x) (13 x + 8) (x + 1) x - ---------------------------------, - 8 (6 x + 5) (1 + 2 x) 6 5 4 3 2 (8821 x + 33939 x + 55191 x + 47329 x + 21906 x + 4980 x + 400) 3 2 3 4 (x + 1) /(24 (6 x + 1) (1 + 2 x) (6 x + 5))]], [1, z, z , z , z ], [(15576 2 2 2 + 47424 z + 306804 x + 2564568 x + 886560 x z + 61128 z + 1081260 x z 3 2 2 3 4 + 30199200 x z + 7831080 x z + 724440 x z + 82576416 x z 3 2 2 3 4 5 + 31049820 x z + 4832280 x z + 291180 x z + 149040864 x z 4 2 3 3 2 4 5 + 75213432 x z + 17178000 x z + 1771560 x z + 71568 x z 3 3 4 4 5 5 + 12197640 x + 43440 z + 37067784 x + 18600 z + 76146912 x + 4896 z 6 6 2 7 8 + 108508776 x + 744 z + 6919200 x z + 107724648 x + 73267200 x 9 10 11 4 6 3 7 + 32602284 x + 8561376 x + 1006992 x + 111456 x z + 4320 x z 6 3 5 4 4 5 3 6 + 35714520 x z + 9836820 x z + 1484784 x z + 108540 x z 2 7 8 2 7 3 6 4 + 2520 x z + 25531200 x z + 15228000 x z + 5257440 x z 5 5 6 5 2 4 3 + 1052352 x z + 180933600 x z + 116479188 x z + 36079680 x z 3 4 7 9 2 8 3 7 4 + 5545680 x z + 48 z + 3922992 x z + 2760480 x z + 1159920 x z 6 5 5 6 10 4 7 2 5 + 295488 x z + 42768 x z + 3058560 x z + 2592 x z + 392112 x z 6 5 3 4 4 3 5 + 10092 z x + 46299960 x z + 9786000 x z + 1053360 x z 2 6 6 2 7 2 7 + 48960 x z + 116106840 x z + 72210420 x z + 146577120 x z 7 8 9 / + 600 x z + 76119840 x z + 22951296 x z) / (24 (6 x + 1) (1 + 2 x) / 6 (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + z + 2) ), (364 + 17856 z + 7180 x 2 2 2 3 + 60290 x + 349440 x z + 53472 z + 993408 x z + 13668912 x z 2 2 3 4 3 2 + 7694478 x z + 1227392 x z + 41168496 x z + 33299808 x z 2 3 4 5 4 2 + 8805172 x z + 842040 x z + 83801328 x z + 90242886 x z 3 3 2 4 5 3 3 + 34536492 x z + 5544390 x z + 347040 x z + 289588 x + 69920 z 4 4 5 5 6 6 + 895844 x + 51000 z + 1894732 x + 22464 z + 2824808 x + 6016 z 2 7 8 9 10 + 2898480 x z + 3002860 x + 2265128 x + 1184504 x + 407958 x 11 12 2 8 4 6 3 7 + 83088 x + 7560 x + 2952 x z + 1639806 x z + 121068 x z 6 3 5 4 4 5 3 6 + 124677892 x z + 50618820 x z + 10947384 x z + 1204952 x z 2 7 8 2 7 3 6 4 + 56988 x z + 80250498 x z + 76650492 x z + 38395170 x z 5 5 6 5 2 4 3 + 10733688 x z + 118329552 x z + 161394912 x z + 82698392 x z 3 4 7 8 9 2 8 3 + 19414500 x z + 912 z + 60 z + 23994576 x z + 26800272 x z 7 4 6 5 5 6 10 + 16104960 x z + 5596128 x z + 1119888 x z + 9018432 x z 4 7 3 8 2 5 6 5 3 + 117936 x z + 4752 x z + 2073960 x z + 86464 z x + 126563080 x z 4 4 3 5 2 6 6 2 + 40115880 x z + 6353928 x z + 462434 x z + 194151378 x z 7 2 10 2 9 3 8 4 + 155881968 x z + 3171960 x z + 4074192 x z + 2873880 x z 7 5 6 6 11 5 7 4 8 + 1205280 x z + 303048 x z + 1052352 x z + 42768 x z + 2592 x z 7 7 8 8 9 + 116419920 x z + 12120 x z + 732 x z + 78486672 x z + 34627536 x z) / 6 / (24 (6 x + 1) (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + z + 2) ), / 2 2 2 (336 + 2864 z + 6112 x + 45836 x + 50712 x z + 25968 z + 484296 x z 3 2 2 3 4 + 1472288 x z + 3796680 x z + 1230848 x z + 3579600 x z 3 2 2 3 4 5 + 16861176 x z + 9171184 x z + 1487736 x z + 5421024 x z 4 2 3 3 2 4 5 + 47808990 x z + 38112672 x z + 10304208 x z + 1024992 x z 3 3 4 4 5 5 + 188020 x + 68480 z + 459794 x + 87280 z + 647584 x + 63936 z 6 6 2 7 8 9 + 328916 x + 28624 z + 368760 x z - 586124 x - 1480240 x - 1637636 x 10 11 12 13 8 5 - 1092672 x - 452376 x - 107730 x - 11340 x + 42768 z x 9 4 2 8 4 6 3 7 + 2592 z x + 68364 x z + 11928210 x z + 1370656 x z 6 3 5 4 4 5 3 6 + 198960448 x z + 133296240 x z + 43658664 x z + 7218008 x z 2 7 8 2 7 3 6 4 + 554064 x z + 73688586 x z + 154999464 x z + 127527536 x z 5 5 6 5 2 4 3 + 53267928 x z + 4734688 x z + 91785276 x z + 99273236 x z 3 4 7 8 9 9 2 + 38951784 x z + 7808 z + 1200 z + 936 x z + 31157820 x z 8 3 7 4 6 5 5 6 + 77715252 x z + 76433268 x z + 39235464 x z + 11268180 x z 10 4 7 3 8 2 5 - 1623000 x z + 1775220 x z + 135972 x z + 6508416 x z 6 5 3 4 4 3 5 + 428632 z x + 171077620 x z + 90003102 x z + 21927360 x z 2 6 6 2 7 2 9 + 2460920 x z + 122615358 x z + 114578940 x z + 80 z 10 2 9 3 8 4 7 5 + 7814178 x z + 22713156 x z + 26150994 x z + 16052040 x z 6 6 11 5 7 4 8 3 9 + 5691006 x z - 483984 x z + 1160964 x z + 124416 x z + 5184 x z 7 2 9 7 8 8 + 1282800 x z + 3528 x z + 108480 x z + 15360 x z - 2056512 x z 9 12 6 7 4 9 - 2796920 x z - 60480 z x + 1193940 z x + 3904092 z x 7 6 3 10 5 8 2 11 / + 303048 z x + 2945160 z x + 2805840 z x + 882252 z x ) / (24 / 6 (6 x + 1) (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + z + 2) ), (784 2 2 2 + 6696 z + 14032 x + 106790 x + 114300 x z + 25752 z + 419748 x z 3 2 2 3 4 + 3338364 x z + 2809944 x z + 1216088 x z + 8931192 x z 3 2 2 3 4 5 + 10610400 x z + 8267488 x z + 2162688 x z + 16788072 x z 4 2 3 3 2 4 5 + 25664199 x z + 32019184 x z + 14386044 x z + 2281536 x z 3 3 4 4 5 5 + 474036 x + 74000 z + 1402711 x + 133440 z + 2988965 x + 146304 z 6 6 2 7 8 + 4838722 x + 100336 z + 815592 x z + 6189932 x + 6404290 x 9 10 11 12 13 + 5362614 x + 3545006 x + 1762840 x + 612367 x + 131229 x 8 5 14 9 4 9 5 8 6 + 1202040 z x + 12978 x + 130896 z x + 42768 z x + 303048 z x 10 4 10 3 2 8 4 6 + 2592 z x + 5616 z x + 696942 x z + 49854497 x z 3 7 6 3 5 4 4 5 + 8547360 x z + 170616010 x z + 206981775 x z + 107049348 x z 3 6 2 7 8 2 7 3 + 26689264 x z + 3154464 x z + 27759375 x z + 150163228 x z 6 4 5 5 6 5 2 + 228316488 x z + 150078096 x z + 23277408 x z + 42471513 x z 4 3 3 4 7 8 9 + 80279946 x z + 53930712 x z + 43632 z + 11760 z + 21180 x z 9 2 8 3 7 4 6 5 + 13503039 x z + 93176654 x z + 171348174 x z + 137824704 x z 5 6 10 4 7 3 8 + 57918091 x z + 7041336 x z + 13271274 x z + 1586964 x z 2 5 6 5 3 4 4 + 14331168 x z + 1481560 z x + 138807804 x z + 128775471 x z 3 5 2 6 6 2 7 2 + 49705632 x z + 8579072 x z + 50023617 x z + 43102569 x z 2 10 9 10 10 2 9 3 + 4248 x z + 1800 z + 120 z + 4841931 x z + 38910956 x z 8 4 7 5 6 6 11 + 83918967 x z + 80253000 x z + 41140213 x z + 2606748 x z 5 7 4 8 3 9 7 + 11958156 x z + 1927986 x z + 153252 x z + 24720168 x z 10 2 9 7 8 8 + 1284 x z + 84384 x z + 603048 x z + 150672 x z + 20669352 x z 9 13 12 6 7 + 13716732 x z + 66528 z x + 609336 z x + 16404021 z x 6 8 4 9 4 10 7 6 + 2818818 z x + 24246891 z x + 3139830 z x + 5833134 z x 7 7 3 10 3 11 5 9 + 1193940 z x + 9844974 z x + 1141812 z x + 3981960 z x 5 8 2 11 2 12 / + 26948700 z x + 1151739 z x + 134190 z x ) / (24 (6 x + 1) / 6 (1 + 2 x) (1 + 3 x) (2 + 3 x) (6 x + 5) (2 x + z + 2) ), (3080 + 27640 z 2 2 2 + 40240 x + 237986 x + 344964 x z + 112200 z + 1337772 x z 3 2 2 3 4 + 6512424 x z + 7169460 x z + 3099440 x z + 14376720 x z 3 2 2 3 4 5 + 22771452 x z + 15786496 x z + 5071272 x z + 21755244 x z 4 2 3 3 2 4 5 + 47525451 x z + 47357264 x z + 25585344 x z + 5977632 x z 3 3 4 4 5 5 + 838852 x + 272000 z + 1944309 x + 452960 z + 3052077 x + 536640 z 6 6 2 7 8 + 3114909 x + 448480 z + 1943208 x z + 1536717 x - 945066 x 9 10 11 12 13 - 2797186 x - 3112424 x - 2235286 x - 1111859 x - 374283 x 15 8 5 14 9 4 9 5 - 7371 x + 12721716 z x - 77175 x + 2098104 z x + 1243116 z x 8 6 10 4 10 5 9 6 + 5975262 z x + 137376 z x + 42768 z x + 303048 z x 8 7 10 3 11 2 11 3 11 4 + 1193940 z x + 172908 z x + 5112 z x + 6048 z x + 2592 z x 2 8 4 6 3 7 6 3 + 4394286 x z + 134466997 x z + 34668592 x z + 114956212 x z 5 4 4 5 3 6 2 7 + 220822275 x z + 184350900 x z + 70793832 x z + 12886200 x z 8 2 7 3 6 4 5 5 + 18891210 x z + 73177616 x z + 229169659 x z + 260482992 x z 6 5 2 4 3 3 4 + 22464400 x z + 68144853 x z + 92737882 x z + 77102248 x z 7 8 9 9 2 8 3 + 259600 z + 101400 z + 238716 x z + 2590116 x z + 30815884 x z 7 4 6 5 5 6 10 + 175313949 x z + 261184596 x z + 172384919 x z - 5289576 x z 4 7 3 8 2 5 6 + 58585866 x z + 10485204 x z + 30165360 x z + 4922272 z x 5 3 4 4 3 5 2 6 + 124025636 x z + 155162859 x z + 91289040 x z + 24282620 x z 6 2 7 2 2 13 2 10 + 67856124 x z + 45789342 x z - 44037 z x + 106344 x z 11 9 10 11 10 2 + 1848 x z + 25480 z + 3720 z + 240 z - 1921272 x z 9 3 8 4 7 5 6 6 + 7837492 x z + 97675173 x z + 183343080 x z + 148837454 x z 11 5 7 4 8 3 9 - 3337704 x z + 63549108 x z + 14950272 x z + 1864328 x z 7 10 2 9 7 8 + 14553984 x z + 31884 x z + 919608 x z + 2755032 x z + 1020360 x z 8 9 13 12 + 3374760 x z - 3958548 x z - 296604 z x - 1292144 z x 7 8 6 7 6 8 4 9 + 2818818 z x + 83221308 z x + 27261369 z x + 37839329 z x 4 10 6 9 7 6 7 7 + 9134325 z x + 3974589 z x + 43115872 z x + 16693632 z x 4 11 3 12 3 10 3 11 + 1031247 z x - 13230 z x + 914196 z x - 28008 z x 5 9 5 10 5 8 14 + 24213816 z x + 3095604 z x + 85973460 z x - 31248 z x 2 11 2 12 / - 1329978 z x - 373653 z x ) / (24 (6 x + 1) (1 + 2 x) (1 + 3 x) / 6 (2 + 3 x) (6 x + 5) (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 -0.1934506232770432920322851794714641493444483317506012453706594604154155372\ -111 829885900402401 10