Answer
$$1$$
Work Step by Step
We have: $0.9999......=0.9+0.09+....=\Sigma_{n=1}^{\infty} 0.9 (0.1)^{n-1}$
This shows a geometric series with common ratio $r=0.1$ and initial term $a=0.9$. We see that $|r| \lt 1$, so the series converges.
Therefore, the sum is: $$0.9999=\dfrac{a}{1-r}\\=\dfrac{0.9}{1-0.1}\\=1$$
