Answer
$$ 0.a_1a_2.......a_n+10^{-n} \times 1$$
Work Step by Step
We have: $0.a_1a_2.......a_n9999......=0.a_1a_2.......a_n+10^{-n} \times 0.9999......$
This shows a geometric series with common ratio $r=0.1$ and with initial term $a=0.9$. We see that $|r|<1$, so the series converges.
Therefore, the sum is:
$$0.9999=\dfrac{a}{1-r}\\=\dfrac{0.9}{1-0.1}\\=1$$ Therefore, the sum is: $$0.a_1a_2.......a_n9999......= 0.a_1a_2.......a_n+10^{-n} \times 1$$
