A good way of looking at it is that as the size of the typical term halves, the number of terms doubles.  The halving is approximate, which allows for the convergence, rather than having a constant value for all n.

It is analagous to use of L'Hopital's rule, where, when the numerator and the denominator of a formula each approach infinity, the limit of the fraction as a whole can be a finite number, dependent on the relationship of the numerator to the denominator.