Chapter 12 Sequences and Series - 12.4 Find Sums of Infinite Geometric Series - 12.4 Exercises - Skill Practice - Page 823: 30

$\dfrac{1}{11}$

Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series. Re-arrange the given series as: $0.090909...=0.09 +0.09 (0.01) +0.09(0.01)^2 +....$ First term $a_1=0.09$ and Common ratio $r=0.01$ The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$ Thus, $S_n=\dfrac{0.09}{1-0.01}$ Hence, $S_n=\dfrac{9}{99}=\dfrac{1}{11}$