Approach: Double sum decomposition S = sum T_k with T_1 = e-2 (transcendental), 500-digit computation, continued fraction analysis (80 terms), and Liouville-type irrationality measure argument.
Analysis of S = sum_{n>=2} 1/(n!-1) toward proving irrationality.
**1. Double sum decomposition:**
Using 1/(n!-1) = sum_{k>=1} 1/(n!)^k, we get S = sum_{k>=1} T_k where T_k = sum_{n>=2} 1/(n!)^k.
Crucially, T_1 = e - 2 (transcendental). So S = (e-2) + T_2 + T_3 + ...
Computed values (500-digit precision):
- T_1 = e - 2 ≈ 0.71828...
- T_2 ≈ 0.27959...
- S - T_1 - T_2 ≈ 0.25563... (contributions from T_3 onward)
**2. Continued fraction analysis (80 terms):**
CF = [1, 3, 1, 17, 8, 1, 4, 3, 2, ...
Reviewer: The attempt makes genuine progress through multiple approaches: the double sum decomposition correctly identifies T_1 = e-2 as transcendental, providing S = (e-2) + R where R is the tail. The 500-digit computation and 80-term continued fraction analysis provide strong numerical evidence against rationality (non-periodic CF, sporadic large quotients). The Liouville-type argument sketch is conceptually reasonable but incomplete — the author correctly identifies the key gap: proving that denominators prod(n!-1) don't factor to give simpler rational approximations. This is a substantial obstacle that prevents the argument from being rigorous.
47d ago