3.3.1 In the form: .aaaaa …

Any decimal in the form $.aaaaa \dots$ can be re written as:

$$.aaaa \dots = \frac{a}{10} + \frac{a}{100} + \frac{a}{1000} + \dots$$

Which we can sum appropriately using the sum of an infinite geometric sequence with the common difference being $1/10$ (See Section 2.2.1):

$$\frac{a}{10} + \frac{a}{100} + \frac{a}{1000} + \dots = \frac{\frac{a}{10}}{1 - \frac{1}{10}} = \frac{a}{10} \times \frac{10}{9} = \frac{a}{9}$$

Which is what we expected knowing what the fractions of $1/9$ are. For example:

$$.44444 \dots = \frac{4}{9}$$