From 95b70729d7cf5a9ffcb5cde3f237fb1ce33413eb Mon Sep 17 00:00:00 2001 From: arc Date: Fri, 3 Oct 2025 11:56:20 -0600 Subject: [PATCH] vault backup: 2025-10-03 11:56:20 --- education/math/MATH1220 (calc II)/Sequences.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/education/math/MATH1220 (calc II)/Sequences.md b/education/math/MATH1220 (calc II)/Sequences.md index 68d1ba7..ac3a401 100644 --- a/education/math/MATH1220 (calc II)/Sequences.md +++ b/education/math/MATH1220 (calc II)/Sequences.md @@ -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. A simple example of an alternating series is: -$$ \sum_{n=1}^\infty (-1)^{n+1}a_n = a_1 - a_2 + a_3 - a_4 $$ \ No newline at end of file +$$ \sum_{n=1}^\infty (-1)^{n+1}a_n = a_1 - a_2 + a_3 - a_4 $$ +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_ \ No newline at end of file