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?:
$$\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}} \cdot \frac{s^2 \sqrt{3}}{4} $$
Best Answer
$$\lim_{n\to\infty} \sum_{r=2}^{n} \frac{3 \cdot 4^{r-2}}{9^{r-1}} \cdot \frac{s^2 \sqrt{3}}{4}=\frac{s^2 \sqrt{3}}{4}\cdot \frac 13\lim_{n\to\infty} \sum_{k=0}^{n-2}\left(\frac 49\right)^k$$ Where $k=r-2$. Now you have a geometric series to sum.