i              |h_i| 3^i              |a_i| 3^i             running S0
   0 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
   1   [2.4472 +/- 7.59e-5]   [3.4472 +/- 7.59e-5]   [4.4472 +/- 7.59e-5]
   2    [3.904 +/- 5.55e-4]    [6.352 +/- 6.48e-4]   [10.799 +/- 5.24e-4]
   3    [4.222 +/- 5.33e-4]    [8.126 +/- 6.98e-4]   [18.925 +/- 8.29e-4]
   4    [3.831 +/- 5.55e-4]     [8.05 +/- 3.70e-3]    [26.98 +/- 3.81e-3]
   5    [2.718 +/- 9.92e-4]     [6.55 +/- 1.64e-3]    [33.53 +/- 5.44e-3]
   6     [1.82 +/- 2.08e-3]     [4.54 +/- 4.27e-3]    [38.06 +/- 8.06e-3]
   7    [0.926 +/- 8.99e-4]     [2.74 +/- 5.58e-3]    [40.81 +/- 6.68e-3]
   8    [0.551 +/- 6.86e-4]     [1.48 +/- 4.59e-3]      [42.3 +/- 0.0213]
   9     [0.17 +/- 4.99e-3]     [0.72 +/- 4.68e-3]    [43.00 +/- 9.38e-3]
  10     [0.15 +/- 2.13e-3]     [0.32 +/- 4.08e-3]      [43.3 +/- 0.0278]
  11     [0.02 +/- 3.02e-3]     [0.13 +/- 2.11e-3]      [43.4 +/- 0.0581]
  12    [0.071 +/- 7.38e-4]     [0.05 +/- 2.48e-3]      [43.5 +/- 0.0146]
  13    [0.054 +/- 6.03e-4]     [0.02 +/- 4.07e-3]      [43.5 +/- 0.0263]
  14    [0.060 +/- 9.38e-4]          [+/- 6.85e-3]      [43.5 +/- 0.0332]
  15    [0.058 +/- 6.90e-4]          [+/- 2.95e-3]      [43.5 +/- 0.0361]
  16     [0.06 +/- 2.18e-3]          [+/- 1.71e-3]      [43.5 +/- 0.0378]

HEAD  sum_{i<=16} |a_i| 3^i <= [43.537789114875633178528999819353684666758941887464441110752810635321998674714098043554561 +/- 2.04e-88]

For the tail i > NMAX, Cauchy on |z| = 4.5 for H(1,z)(1-z/3):
  |a_i| <= M0 4.5^-i, so sum_{i>NMAX} |a_i| 3^i <= M0 (2/3)^(NMAX+1) / (1/3)
  -> run s0_cauchy.py for a certified M0