>> But "intuitively" 0.99999... is not the same as 1.
According to whom? It seems intuitive to me.
>> It is the sum (9/10^n) as n goes to infinity.
>> The limit of this summation is 1, but left on its own, it never >> quite gets there on its own (always 1/10^n away).
I don't know what you mean by "left on its own". The summation is 1. Any partial sum is less than 1, in fact (as you state) always 1/10^n away for any finite n. But the complete sum leaves out no finite n. So it is never 1/10^n away for any n.
>> Obviously (I think) this goes against "common sense," and, while >> you are probably very correct... it is a little tough to swallow.
Whose common sense? It makes sense to me. And there's no probably about it.
There are a number or counter-intuitive results in mathematics, but that doesn't make them false. "Common sense" would not tell you that folding a piece of paper in half, then in half again, and continue 50 times, that the thickness of the final wad would exceed the distance between the Earth and the Sun. But it does. (Do the math: 2^50 * thickness of paper.)