The way '...' is used here is perfectly consistent with being defined as a geometric series where the ratio between successive elements is 1/10 and the start te

The way '...' is used here is perfectly consistent with being defined as a geometric series where the ratio between successive elements is 1/10 and the start term is the final digit. Geometric series always converge when the absolute value of the ratio of less than 1. I should note that when I learned about rational & irrational numbers in elementary school (I think third or fourth grade), we used a "bar" notation where we'd put a bar over the last digits in a decimal expression that repeated forever (i.e. it corresponded exactly to a geometric series with r = (1 / 10)^k where k is the number of digits under the bar, though we didn't know about that at the time). Our teachers explained that the difference between a rational and irrational number was that there would be no pattern you could ever find in an irrational number that would allow us to use the bar, which is surprisingly accurate for grade school arithmetic.