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education/math/MATH1220 (calc II)/Sequences.md

@@ -40,7 +40,7 @@ Remember, L'Hospital's rule states that:
 # Series
 Vocabulary: A **series** is another name for a sum of numbers. 
 
-## Properties
+## The Limit Test
 You can break a series into *partial sums*:
 $$\sum_{n=1}^\infty a_n = a_1 + 1_2 + a_3 + ...$$
 Given the above series, we can define the following:
@@ -52,10 +52,16 @@ Given the above series, we can define the following:
 and say that the sum converges to $L$. 
 
 ## Examples
-> Prove that $\sum_{n = 1}^\infty a_n = L$
+> Prove that $\sum_{n = 1}^\infty \frac{1}{2^n} = 1$
 - $S_1 = \frac{1}{2}$
 - $S_2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$ 
 - $S_3 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8}$ 
 - $S_n = \frac{2^n - 1}{2^n}$
 So:
-$$ \sum_{n=1}^\infty \frac{1}{2^n} = \lim_{n \to \infty}S_n = \lim_{n \to \infty} (1 - \frac{}{}$$
+$$ \sum_{n=1}^\infty \frac{1}{2^n} = \lim_{n \to \infty}S_n = \lim_{n \to \infty} (1 - \frac{1}{2^n}) = 1$$
+
+> Using the limit test, determine whether the series $\sum_{n = 1}^\infty n$  converges or diverges
+- $S_1 = 1$
+- $S_2 = 1 + 2 = 3$
+- $S_n = \frac{n(n+1)}{2}$ 
+So: