Question 1
Given
$$S=1+\frac1{1+\frac13}+\frac1{1+\frac13+\frac16}+\cdots+\frac1{1+\frac13+\frac16+\cdots+\frac1{1993006}},$$
where the denominators are partial sums of the reciprocals of triangular numbers $k=n(n+1)/2$ for $n=1,2,\ldots,1996$, prove that $S\gt 1001$.
$$S=1+\frac1{1+\frac13}+\frac1{1+\frac13+\frac16}+\cdots+\frac1{1+\frac13+\frac16+\cdots+\frac1{1993006}},$$
where the denominators are partial sums of the reciprocals of triangular numbers $k=n(n+1)/2$ for $n=1,2,\ldots,1996$, prove that $S\gt 1001$.