Going in the opposite direction:

1,2 the odds are 1-1, zero!

For 1,2,3, 1-2/3*1/2=4/6; 4/6-3/4=-1/12;

for 1,2,3,4, 1-2/4*1/3=5/6; 5/6-3/4=1/12;

1...5, 1-3/5*2/4=7/10; 7/10-3/4=-1/20

1...6, 1-3/6*2/5=8/10; 8/10-3/4=1/20

1...7, 1-4/7*3/6=10/14; 10/14-3/4=-1/28;

1...8, 1-4/8*3/7=11/14; 11/14-3/4=1/28

1...9, 1-5/9*4/8=13/18; 13/18-3/4=-1/36;

1...10, 1-5/10*4/9=14/18; 14/18-3/4=1/36

etc., i.e., the pattern is obvious.

The odds for odd numbers is:(3/4)-1/(4*odd);

and the odds for even (odd plus one) is:

(3/4)+1/(4*odd).

The upward and downward convergence on 3/4 is self evident.

[BTW, the answer to the puzzle is 3/4+1/(4*2024), a hair over 3/4.]