Length removed during the construction of the Cantor set

mark

Use a geometric series to show that the total length of the intervals removed during the construction of the Cantor set is 1.

theoldernoah

[dmath] \sum_{n=0}^{\infty} 2^n \cdot \frac{1}{3^{n+1}} = \frac{1}{3} \sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n = \frac{1}{3} \cdot \frac{1}{1-\frac{2}{3}} = 1. [/dmath]