Sequence and Series

From merely the first few terms, it is impossible to know all the terms of a sequence. For example, a sequence whose first five terms are \(1, 2, 3, 4, 5\) could be

1. the sequence of all positive integers; or
2. the sequence \(1, 2, 3, 4, 5, 0, 0, 0, \dots\) whose first five terms are \(1, 2, 3, 4, 5\) but the rest of the terms are all \(0\)'s.