Does the infinite series 9/10^n really sum to 1?

Yes. The series $$\sum_{n=1}^{\infty}\frac{9}{10^n}$$ is a geometric series. The first term is: $$a=\frac{9}{10},$$ and the common ratio is: $$r=\frac{1}{10}.$$ For a geometric series with $|r|<1$: $$a+ar+ar^2+\cdots=\frac{a}{1-r}.$$ So: $$\sum_{n=1}^{\infty}\frac{9}{10^n} =\frac{9/10}{1-1/10} =\frac{9/10}{9/10} =1.$$ This is the same idea behind: $$0.999\ldots=1.$$ The partial sums $0.9, 0.99, 0.999,\ldots$ approach 1, and the infinite decimal is defined as that limit. So the phrase "it only approaches 1" applies to the finite partial sums. The completed infinite series is the limit itself, and that limit is exactly 1.