final update on 4121

pages/Math4121/Math4121_L19.md

@@ -22,7 +22,23 @@ $$
 
 However, the quarter cantor set removes $\frac{3^{n-1}}{4^n}$ of the total "content", and the total length removed after infinitely many steps is:
 
-_skip this part, some error occurred._
+Every time we remove $\frac{1}{4^n}$ of the remaining intervals. So on each layer, we remove $\frac{2^{n-1}}{4^n}$ of the total "content".
+
+So the total length removed is:
+
+$$
+\begin{aligned}
+1-\frac{1}{4}-\frac{2}{4^2}-\frac{2^2}{4^3}-\cdots&=1-\frac{1}{4}\sum_{n=0}^{\infty} \left(\frac{2}{4}\right)^n\\
+&=1-\frac{1}{4}\cdot\frac{1}{1-\frac{2}{4}}\\
+&=1-\frac{1}{4}\cdot\frac{4}{2}\\
+&=1-\frac{1}{2}\\
+&=\frac{1}{2}
+\end{aligned}
+$$
+
+#### Generalized Cantor set (SVC(n))
+
+The outer content of $SVC(n)$ is $\frac{n-3}{n-2}$.
 
 #### Monotonicity of outer content