From a theoretical point of view, if that occurs only for a finite number of indices, one could easily agree to consider the sequence  restricted to indices   with a sufficiently large . From a practical point of view, one does in general rather consider only the first few terms of the sequence, which usually provide the needed precision.
Theoretically, if the sequence has a finite number of indices, we can consider only those with a large enough value. Practically speaking, however, we often focus on the first few terms to achieve sufficient precision.