Sorry I'm not sure what you mean.
Do you mean when can you use arithmetic or geometric series to solve problems?
Or when can you use sum to infinity formula?
Well there's the good old everyday common occurance of where you drop a bouncy ball off a cliff and it every bounce it bounces half the height of the previous one. Of course this ball never stop bouncing so we want to know how much distance was covered by the ball.
Like I said, very common.