The acceleration formula for Zeta(, 4, ) is Theorem. Let the column-vector of functions of, x T a(x):=, [a[0](x), a[1](x), a[2](x)] , be defined T by the initial conditions:, a(0) = [1, 0, 0] and a(x+1)=A(x)a(x), where A(x) is the, 3, by , 3, matrix 2 3 (23 x + 76 x + 63) (x + 1) [[----------------------------, 4 32 (4 x + 3) (2 x + 3) 4 3 2 3 (150 x + 670 x + 1041 x + 652 x + 141) (x + 1) - --------------------------------------------------, 4 64 (1 + 2 x) (4 x + 3) (2 x + 3) 6 5 4 3 2 3 (900 x + 4870 x + 10211 x + 10534 x + 5629 x + 1470 x + 144) (x + 1) / 4 / (64 (4 x + 1) (1 + 2 x) (4 x + 3) (2 x + 3) )], [ / 3 3 2 3 5 (3 x + 5) (x + 1) (94 x + 266 x + 215 x + 53) (x + 1) -----------------------, - --------------------------------------, 4 4 16 (4 x + 3) (2 x + 3) 32 (1 + 2 x) (4 x + 3) (2 x + 3) 5 4 3 2 3 (540 x + 2040 x + 2778 x + 1732 x + 495 x + 51) (x + 1) ------------------------------------------------------------], [ 4 32 (4 x + 1) (1 + 2 x) (4 x + 3) (2 x + 3) 3 3 5 (x + 1) 5 (3 x + 2) (x + 1) -----------------------, - -----------------------, 4 4 16 (4 x + 3) (2 x + 3) 32 (4 x + 3) (2 x + 3) 3 2 3 (82 x + 133 x + 65 x + 9) (x + 1) ------------------------------------]] 4 32 (4 x + 1) (4 x + 3) (2 x + 3) T In other words, a(x)=A(x-1)A(x-2)...A(0) times, [1, 0, 0] Let 3 2 a[0](x) (135 x + 103) a[1](x) (10 x + 52 x + 67 x + 25) b(x) = 1/32 --------------------- + 1/64 ----------------------------------- 4 x + 3 (1 + 2 x) (4 x + 3) 5 4 3 2 a[2](x) (60 x + 150 x - x - 265 x - 234 x - 60) - 1/64 --------------------------------------------------- (4 x + 1) (1 + 2 x) (4 x + 3) We have the following acceleration formula (with convergence-rate, 0.0009765625000, ) infinity infinity ----- ----- \ 1 \ ) -------- = ) b(x) / 4 / ----- (z + 1) ----- z = 0 x = 0 Proof: The following is a Markov-WZ pair 4 2 3 ((z + x)!) (23 x + 76 x + 63) (x + 1) [-----------------, [[----------------------------, 4 2 (4 x + 3) ((2 x + z + 1)!) 4 3 2 3 (150 x + 670 x + 1041 x + 652 x + 141) (x + 1) - --------------------------------------------------, 4 (1 + 2 x) (4 x + 3) 6 5 4 3 2 3 (900 x + 4870 x + 10211 x + 10534 x + 5629 x + 1470 x + 144) (x + 1) / 3 5 (3 x + 5) (x + 1) (4 (4 x + 1) (1 + 2 x) (4 x + 3))], [--------------------, 4 x + 3 3 2 3 (94 x + 266 x + 215 x + 53) (x + 1) - --------------------------------------, 2 (1 + 2 x) (4 x + 3) 5 4 3 2 3 (540 x + 2040 x + 2778 x + 1732 x + 495 x + 51) (x + 1) ------------------------------------------------------------], 2 (4 x + 1) (1 + 2 x) (4 x + 3) 3 3 3 2 3 5 (x + 1) 5 (3 x + 2) (x + 1) (82 x + 133 x + 65 x + 9) (x + 1) [----------, - --------------------, ------------------------------------]] 4 x + 3 2 (4 x + 3) 2 (4 x + 1) (4 x + 3) 2 2 3 4 , [1, z, z ], [(206 + 444 z + 2330 x + 10528 x + 25092 x + 34478 x 5 6 7 2 2 + 27538 x + 11908 x + 2160 x + 4568 x z + 388 z + 3588 x z 3 2 2 3 2 3 4 + 35680 x z + 12020 x z + 1440 x z + 18024 x z + 176 z + 42 z 5 3 3 3 4 5 6 2 4 + 4 z + 4320 z x + 1664 z x + 20712 z x + 4576 z x + 13752 z x 2 5 2 3 4 4 4 2 4 3 + 3872 z x + 18700 z x + 37916 z x + 298 z x + 612 z x + 368 z x 3 2 5 5 2 / + 3920 z x + 24 z x + 32 z x ) / (4 (4 x + 1) (1 + 2 x) (4 x + 3) / 4 2 3 4 (2 x + z + 2) ), (25 + 318 z + 267 x + 1138 x + 2600 x + 3525 x 5 6 7 8 2 2 + 2915 x + 1432 x + 378 x + 40 x + 3408 x z + 644 z + 6278 x z 3 2 2 3 2 3 4 + 32552 x z + 23254 x z + 4928 x z + 14488 x z + 568 z + 261 z 5 6 6 6 2 3 3 3 4 + 62 z + 6 z + 32 z x + 32 z x + 22000 z x + 15152 z x 7 5 6 2 4 2 5 + 2320 z x + 32200 z x + 13316 z x + 43646 z x + 22732 z x 2 6 5 3 2 3 4 4 + 4816 z x + 368 z x + 43350 z x + 42358 z x + 1975 z x 4 2 4 3 4 4 3 2 5 + 4820 z x + 4810 z x + 1704 z x + 15240 z x + 400 z x 5 2 3 5 / + 700 z x + 4032 z x ) / (4 (4 x + 1) (1 + 2 x) (4 x + 3) / 4 2 3 4 5 (2 x + z + 2) ), (60 + 354 z + 474 x + 1561 x + 2705 x + 2440 x + 640 x 6 7 8 2 2 3 - 871 x - 959 x - 390 x + 2584 x z + 1074 z + 8076 x z + 12480 x z 2 2 3 2 3 4 9 + 25998 x z + 11248 x z + 7816 x z + 1584 z + 1248 z - 60 x 5 6 7 7 2 7 6 6 3 + 546 z + 126 z + 12 z + 32 z x + 40 z x + 552 z x + 368 z x 6 2 3 3 3 4 8 7 5 + 788 z x + 53040 z x + 47576 z x - 200 z x - 824 z x + 4304 z x 2 7 6 2 4 2 5 2 6 + 1960 z x - 276 z x + 50488 z x + 33126 z x + 12224 z x 5 3 2 3 4 4 3 6 + 5200 z x + 46622 z x + 10850 z x + 7898 z x + 4576 z x 4 2 4 3 4 4 4 5 3 2 + 19965 z x + 25185 z x + 15842 z x + 3972 z x + 33384 z x 5 4 5 5 2 3 5 / + 1704 z x + 2944 z x + 5896 z x + 22832 z x ) / (4 (4 x + 1) / 4 (1 + 2 x) (4 x + 3) (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 -0.9321439656095722205297890085495627239992875449526374431902381797209569109\ -93 579189394223552631188652867565109 10