How do I write a repeating decimal as an infinite geometric series?
It helps to think about what digits in a base 10 system represent. The digits to the left of a decimal point represents an increasing power of 10 (starting from 0), whereas those to the right of a decimal point represent a decreasing power of 10 (starting from -1).
Now we can figure out how to write a repeating decimal as an infinite sum. An infinite geometric series is a series of the form
From the properties of decimal digits noted above, we can see that the common ratio will be a negative power of 10. What the power is will be determined by how many digits are repeating, and what
or, in general,
Finally, note that a repeating decimal may only start repeating after a finite number of initial decimal digits. In that case, we simply add the finite digits to the series separately. For example:
(Note that it is the digits that are repeating which determine
Thus, we get the most general form: