update notations and fix typos

pages/Math4111/Math4111_L20.md

@@ -114,7 +114,7 @@ Therefore, $e\leq \liminf_{n\to\infty} t_n\leq \limsup_{n\to\infty} t_n\leq e$.
 
 So $\lim_{n\to\infty} t_n$ exists and $\lim_{n\to\infty} t_n = e$.
 
-EOP
+QED
 
 #### Theorem 3.32
 
@@ -156,7 +156,7 @@ $$
 
 Contradiction.
 
-EOP
+QED
 
 ### The root and ratio tests
 
@@ -190,7 +190,7 @@ Thus $a_n\not\to 0$, $\sum_{n=0}^{\infty} a_n$ diverges.
 
 (c) $\sum_{n=0}^{\infty} \frac{1}{n}$ and $\sum_{n=0}^{\infty} \frac{1}{n^2}$ both have $\alpha = 1$. but the first diverges and the second converges.
 
-EOP
+QED
 
 #### Theorem 3.34 (Ratio test)
 
@@ -232,7 +232,7 @@ i.e. $\forall n\geq N, |a_n| < \beta^{n-N}|a_N|=\beta^n(\beta^{-N}|a_N|)$.
 
 Since $\sum_{n=N}^{\infty} \beta^n$ converges, by comparison test, $\sum_{n=0}^{\infty} a_n$ converges.
 
-EOP
+QED
 
 We will skip **Theorem 3.37**. One implication is that if ratio test can be applied, then root test can be applied.
 
@@ -267,4 +267,4 @@ $$
 
 By root test, the series converges absolutely for all $z\in\mathbb{C}$ with $|z| < R$.
 
-EOP
+QED