The reason this is a "paradox" is that the logic seems irrefutable although our common sense tells us otherwise. It isn't a true paradox because it isn't a logical contradiction but rather the reasoning seems to go against common sense.

To actually show why this isn't a true paradox involves understanding infinite series. We can build an infinite series out of the sequence 1, 1/2, 1/4, 1/8... Now every term of the sequence is positive. "Logically" adding all the terms would result in "infinity" as there are an infinite number of positive numbers added together.

It actually isn't obvious to a person not familiar with infinite series, why this "logic" isn't true.