@James:

If I recall my Newton correctly, the only way to take this "sum of an infinite series" business consistently is to interpret it as shorthand for the limit of an infinite series. (Cf. Newton's Principia Mathematica, Lemma 2. The infinitesimally wide parallelograms are dubitably real, but the area under the curve between the sets of parallelograms is clearly a real, definite area.)

@Benoit:

Why shouldn't we take 1.9999... as just another, needlessly complicated (if there's no justifying context) way of writing "2"? Just as I could conceivably count "1, 2, 3, 4, d(5x)/dx, 6, 7" if I were a crazy person.