Sum of a geometric series $\sum_0^\infty \frac{1}{2^{1+2n}}$ $$\sum_0^\infty \frac{1}{2^{1+3n}}$$
So maybe I have written the sequence incorrectly, but how do I apply the $\frac{1}{1 - r}$ formula for summing a geometric sequence to this? When I do it I get something over one which is wrong because this is suppose to model a percentage of something.

Write $$S=\sum_{n=0}^{\infty}\frac{1}{2^{1+2n}}=\frac{1}{2}+\frac{1}{2^3}+\frac{1}{2^5}+\cdots$$ Then note that $$\frac{S}{2^2}=S-\frac{1}{2}\Rightarrow S=\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$$
