diff --git a/education/math/MATH1220 (calc II)/Sequences.md b/education/math/MATH1220 (calc II)/Sequences.md index 6fbc43a..b722240 100644 --- a/education/math/MATH1220 (calc II)/Sequences.md +++ b/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: -$$ \sum_{n=1}^\infty \frac{1}{2^n} = \lim_{n \to \infty}S_n = \lim_{n \to \infty} (1 - \frac{}{}$$ \ No newline at end of file +$$ \sum_{n=1}^\infty \frac{1}{2^n} = \lim_{n \to \infty}S_n = \lim_{n \to \infty} (1 - \frac{1}{2^n}) = 1$$ + +> 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: \ No newline at end of file