> 1 + 1/2 + 1/4 + 1/8 ... = 2

It's not necessarily clear that this statement requires infinity, if you're willing to treat "... =" as a shorthand. You might prefer something like "1 + 1/2 + 1/4 + 1/8 ... -> 2" if it's more clear, where "->" means something like "gets as close as you like without ever getting further away than that", but really the "=" sign is already overloaded in all sorts of subtly different ways anyway, so there's not really any trouble using it here. Almost any rigorous definition you can write down of exactly what that statement means would not rely on the use of infinity.