A series is sometimes an excellent way to represent a physical phenomenon. Population growth is an example most of us are familiar with. Zeno’s paradoxes are pretty well known too. The sum of the series is, in many cases, what we are really interested in. If the series has infinite terms though, we are (like Zeno’s Achilles) more interested in what the sum will converge to.
In real life we settle for an approximate sum rather than insist on the precise sum. That is, we’ll keep adding terms until the difference between successive iterations is less than whatever we think is acceptable.
This is a convenient representation of convergence, particularly for CFD users who commonly monitor the convergence of residuals of energy and velocity.
Convergence is a very elegant subject full of nuances. This is as good a place to start as any, though you should really refer to Piskunov for the truth, the whole truth and nothing but the truth.
