I'm not sure if this will help or not, but a proof for the "ratio" test of series seems to be similar to what you're doing. Maybe worth a shot. Basically using the logic that having the n+1 term divided by the nth term is the same as having the nth term divided by the n-1 term. Here is a link to the proof for the ratio test, although I haven't yet checked it out myself. Again, I'm still in highschool, so I'm not quite sure if this will help or not.