One of the coolest parts about a geometric series is the ability to rationalize and control patterns that emerge from it. Althought these recurring patterns can often seem arbitrary, they can actually be represented in a concise way using basic formulae of geometric series.

For example, the pattern $0.\dot{5}\dot{0}$. Without the dot notation, this works out to be $0.5050505050$ ad infinitum, which is annoying when we need a form to work with. How can we express this recurring decimal in a fraction? Well, when we think about it, a recurring decimal is just a geometric series:

Each term is related to its predecessor by a common ratio, $r$. To find this ratio, we divide a term by its predecessor:

As the series progresses, the next term is equated by dividing the current term by 100. We know that the first term, $a$, is 0.50. To find the total sum of a geometric series to $n$ terms:

But we want to find the sum of the series to an infinite amount of terms. How do we do this?

We say that $S_n$ tends towards a limiting value as $n$ approaches infinity. So, in our case, each term will approximate to zero for large values of $n$, i.e. $\frac{n}{100} \to 0$ as $n \to \infty$. Thus:

Which works out as:

$$S_{\infty} = \frac{a}{(1 - r)}$$

However, $S_{\infty}$ only exists for infinite geometric series in which $% $. This is called a convergent series. So when we apply this equation to our number:

Cool! The recurring decimal $0.505050$ can also be represented as $\frac{50}{99}$.

Let’s try another: $0.\dot{3}6\dot{0}$ ($0.360360360$):

Awesome!