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math119: add last convergences

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eggy 2023-03-31 11:53:59 -04:00
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@ -756,6 +756,24 @@ For two series $\sum a_n$ and $\sum b_n$ where **all terms are positive**, if $a
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. 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.
### Ratio tests
The **ratio test** is applicable if the $L$ exists or is infinity:
$$L+\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$$
- $L<1$ implies the function converges absolutely
- $L>1$ implies the function diverges
- $L=1$ is inconclusive
It is useful if a constant is raised to the power of $n$ or if a factorial is present.
The **root test** has the same analysis but with a different limit:
$$L=\lim_{n\to\infty}\sqrt[n]{|a_n|}$$
It is useful for functions of the form $f(x)^{g(x)}$.
### Alternating series ### 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. 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.
@ -771,3 +789,5 @@ $\sum a_n$ converges **absolutely** only if $\sum |a_n|$ converges.
An absolutely converging series also has its regular form converge. 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 $b$. 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 $b$.