If that is
the reasoning behind the paradox, it's no wonder you get a paradox! One certainly can find the exact sum of an infinite series of numbers. For example the sum 1+ 1/3+ 1/9+ 1/27+ ... is exactly
1.5. That is not a "finite number that it may approach". I'm not certain what you mean by "it" here. If you mean the sum, it is not approaching anything, it is
1.5. If you mean the sequence of partial sums (which is what many people mean when they talk about something like this), there is no "may" that sequence is approaching that number and so sum is, by definition, 1.5.