Chapter 12 Sequences and Series - 12.4 Find Sums of Infinite Geometric Series - 12.4 Exercises - Skill Practice - Page 823: 33

$1$

Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series. Re-arrange the given series as: $0.9999...=0.9 +0.09 (0.1) +0.9(0.1)^2 +....$ First term $a_1=0.9$ and Common ratio $r=0.1$ The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$ Thus, $S_n=\dfrac{0.9}{1-0.1}$ Hence, $S_n=\dfrac{0.9}{0.9}=1$