What is the sum of (2 to the n-1)/(3 to the n) where n goes from 1 to infinity?
In other words what is the sum of 1/3 + 2/9 + 4/27 + 8/81 ...
Hello,
use the sum formula for infinite geometric series:
$\displaystyle \sum_1^{\infty}\left( \frac{2^{n-1}}{3^n} \right) = \sum_1^{\infty}\left( \frac{2^{n-1}}{3^n} \right) = \sum_1^{\infty}\left( \frac{1}{2} \cdot \left(\frac{2}{3} \right)^n \right) = \frac{\frac{1}{3}}{1-\frac{2}{3}} = 1$
Thanks earboth.
So what you are saying is,
if I have a pie and cut it into three pieces and ate one of them, I would have two one-third pieces left. Then if I cut each of those one-third pieces into three pieces and ate one, I would have four one-ninth pieces. And if I continue cutting each piece into threes and eating one of them, I will eventually eat the entire pie.
Right?????