Sorry, that was me above and I got cut off mid sentence. The mistake is getting too hung up on the method of getting the answer rather than the answer itself. That way, instead of concluding that the terminating and recurring numbers are in some sense distinct, you start wondering if the problem lies in the method of adding the digits rather than the numbers themselves. Obviously enough, mathematicians don't handle infinite series by adding all the terms together explicitly since by definition that is an infinite process. Exactly how the do it varies but a common way is to prove the sum of the series must be X or more and simultaneously X or less. Hence you can work out the sum without actually doing the calculation.

And to get back onto representation again: there's no need to stick to a base-10 notation. If you use a base 7, for example, a number like 2/7 that is recurring in base 10 'decimals' is not in base 7 'septimals'. So a suggestion there are distinct types of number depending on whether they recur of not is clearly an artifact of the chosen notation, not of the number