You still seem confused. What you have is not a converging series, period. You just have a list of successively-closer ratios. A series (in math) is the sum of the terms in a sequence (which itself is a discrete function). You don't have a discrete function, and you definitely don't have the sum of anything.
He does have a converging sequence, though. Many people don't understand the technical (but obvious to mathematicians) distinction between sequence and series.
