You must be thinking of
Code:
1/2 + 1/4 + 1/8 + 1/16 + 1/32...
which approaches 1. The arithmetic variation
Code:
1/2 + 1/3 + 1/4 + 1/5  + 1/6..
is not only bigger at each stage, but the margin by which it is bigger grows as well, thus resulting in what you originally stated.