We know all we need to know about geometric series. As a nifty bonus, we can use geometric series to better understand infinite repeating decimals. This is something you can rub in your former math teachers' faces.
Does 0.99999... equal 1?
First we have to figure out what 0.99999... means. If we break it up by individual decimal places,
0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0 .00009 + ....
Convert each term to a fraction. Now
This is an infinite geometric series with and . The sum of the series is
We've just proved that
.99999... = 1.
Haha! Take that 8th-grade math teacher that took off 5 points from your test when you wrote '1' instead of '0.9999...'.
The same trick can be used to turn other infinite repeating decimals back into rational numbers.
when written as a fraction?
as a rational number.