Chapter 11 - Further Topics in Algebra - Summary Exercises on Sequences and Series - Exercises - Page 1038: 26

$ 0.3+0.03+0.003...= \frac{1}{3}$

We have $0.333...=0.3+0.03+0.003...$ and it is a geometric sequence with $a_1=0.3, r=\frac{1}{10}$. The sum is $0.333...=0.3+0.03+0.003...=\sum_{i=1}^{\infty}0.3(\frac{1}{10})^{i-1}=0.3\times\frac{1}{1-\frac{1}{10}}=\frac{1}{3}$