math119: finalise convergence tests

docs/1b/math119.md

@@ -750,3 +750,24 @@ $$\sum^\infty_{n=1}\frac{1}{n^p}$$
 
 converges if and only if $p>1$.
 
+### Comparison test
+
+For two series $\sum a_n$ and $\sum b_n$ where **all terms are positive**, if $a_n\leq b_n$ for all $n$, either both converge or both diverge.
+
+The **limit comparison test** has the same requirements, but if $L=\lim_{n\to\infty}\frac{a_n}{b_n}$ such that $0<L<\infty$, either both converge or both diverge.
+
+### Alternating series
+
+If the absolute value of all terms $b_k$ continuously decreases and $\lim_{k\to b_k}=0$, the alternating function $\sum^\infty_{k=0}(-1)^kb_k$ converges.
+
+The **alternating series estimation theorem** places an upper bound on the error of a partial sum. If the series passes the alternate series test, $S_n$ is the $n$th partial sum, $S$ is the sum of the series, and $b_k$ is the $k$th term:
+
+$$|S-S_n|\leq b_{n+1}$$
+
+### Conditional convergence
+
+$\sum a_n$ converges **absolutely** only if $\sum |a_n|$ converges.
+
+An absolutely converging series also has its regular form converge.
+
+A series converges **conditionally** if it converges but not absolutely. This indicates that it is possible for all $b\in\mathbb R$ to rearrange $\sum a_n$ to cause it to converge to $b$.