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education/math/MATH1220 (calc II)/Sequences.md
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@@ -93,4 +93,8 @@ The divergence test only tells us that if the limit does not equal zero, then th
Sometimes a series is not continually positive for the entire series, meaning most tests on series do not apply. To get around this, you can split the series into two or more parts. A finite negative number + infinity is still infinity, and a finite negative number + a finite number is still a finite number. Sometimes a series is not continually positive for the entire series, meaning most tests on series do not apply. To get around this, you can split the series into two or more parts. A finite negative number + infinity is still infinity, and a finite negative number + a finite number is still a finite number.
A simple example of an alternating series is: A simple example of an alternating series is:
$$ \sum_{n=1}^\infty (-1)^{n+1}a_n = a_1 - a_2 + a_3 - a_4 $$ $$ \sum_{n=1}^\infty (-1)^{n+1}a_n = a_1 - a_2 + a_3 - a_4 $$
The above series converges if all three of the following hold true:
- $a_n > 0$
- Series decreases: $a_n \ge a_{n+1}$ for all $n$
- $\lim_{n\to\infty} a_n = 0$ as_