Is it possible to add all of the terms in an infinite sequence and get a number?

In order to add an infinite list of nonzero numbers and get a finite result, what must be true about most of the numbers in that list?

If we add an infinite list of numbers and the sum diverges (either by oscillating or by approaching \(\pm \infty\)), will the series still diverge if the first 100,000,000 terms are removed?

Does the order in which we add infinitely many numbers affect the value of a sum of infinitely many numbers?