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education/math/MATH1220 (calc II)/Sequences.md
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@@ -42,7 +42,7 @@ Vocabulary: A **series** is another name for a sum of numbers.
## The Limit Test ## The Limit Test
You can break a series into *partial sums*: You can break a series into *partial sums*:
$$\sum_{n=1}^\infty a_n = a_1 + 1_2 + a_3 + ...$$ $$\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + ...$$
Given the above series, we can define the following: Given the above series, we can define the following:
- $S_1 = a_1 = \sum_{i=1}^\infty a_i$ - $S_1 = a_1 = \sum_{i=1}^\infty a_i$
- $S_2 = a_1 + a_2 = \sum_{i=1}^2 a_i$ - $S_2 = a_1 + a_2 = \sum_{i=1}^2 a_i$
@@ -105,5 +105,11 @@ A sequence $a_n$ *converges absolutely* if $\sum |a_n|$ converges
Then if the *series converges* absolutely then the sum converges. Then if the *series converges* absolutely then the sum converges.
## Examples ## Examples
> Does the series $\sum_{n=1}^\infty (\frac{(-1)^n}{n^2}$ conditionally converge, absolutely converge, or diverge?
1. $\sum_{n=1}^\infty|\frac{(-1)^n}{n^2}| = \sum_{n=1}^\infty \frac{1}{n^2}$
2. We know that $\sum_{n=1}^\infty \frac{1}{n^2}$ converges by the p test, so the series absolutely converges.
> Does the series $\sum_{n=1}^\infty (\frac{(-1)^n}{n+5}$ conditionally converge, absolutely converge, or diverge? > Does the series $\sum_{n=1}^\infty (\frac{(-1)^n}{n+5}$ conditionally converge, absolutely converge, or diverge?
1. $\sum_{n=1}^\infty|\frac{(-1)^n}{n+5}| = \sum_{n=1}^\infty \frac{1}{n+5}$ 1. Consider $\sum_{n=1}^\infty|\frac{(-1)^n}{n+5}| = \sum_{n=1}^\infty \frac{1}{n+5}$.
2. The above series can be compared to $\frac{1}{n}$ with the limit comparison test.