Chapter 12 Sequences and Series - 12.4 Find Sums of Infinite Geometric Series - 12.4 Exercises - Skill Practice - Page 823: 27

$\dfrac{625}{999}$

Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series. Re-arrange the given series as: $0.625625625...=0.625 +0.625 (0.001) +0.625(0.001)^2 +....$ First term $a_1=0.625$ and Common ratio $r=0.001$ The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$ Thus, $S_n=\dfrac{0.625}{1-0.001}$ Hence, $S_n=\dfrac{625}{999}$