Finding the limit of the area of a Koch Snowflake this is my first question for this site and I made this account specifically for help with the following topic.
I am doing a research presentation on the Koch Snowflake, specifically, the area. 
So far, I have been attempting to generalize a formula for finding the area of the snowflake at n iterations, and I am now trying to find the limit as n tends toward infinity. 
So, basically, what is the limit for the following?: 
$$\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}}  \cdot \frac{s^2 \sqrt{3}}{4} $$

The following forms a GP with $a=\frac{1}{9}$ and $r=\frac{4}{9}$.
$$\begin{align}
&\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}}  \cdot \frac{s^2 \sqrt{3}}{4}\\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{4}\lim_{n\to\infty} \sum_{r=2}^{n} \frac{ 4^{r-2}}{9^{r-1}} \\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{4}\cdot\frac{\frac{1}{9}}{1-\frac{4}{9}}\\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{4}\cdot\frac{1}{5}\\
=\ &\frac{ 3\sqrt{3}\cdot s^2}{20}
\end{align}\\
$$
