Which geometric series represents 0. as a fraction?

Here the first term of this series is 0.4.  The next is 0.1(0.4), or 0.04.  The next is (0.1)(0.1)(0.4), or 0.004.  Thus, the common ratio, r, is 0.1.

The sum of this series is    a/(1-r), where a is the first term and r is the common ratio.

Subst. the known values of a and r,    the sum of this series is 0.4/(1-0.1), or

0.4        4
= ---
 0.9       9

Divide 4 by 9 on your calculator.  You'll find that the result is the repeating fraction 0.44444

Thus, 4/9 is equivalent to 0.4444