Convergence Tests
TODO: Mention absolute vs conditional convergence, possibly make table of contents
Important Note: All finite series converge, as long as each individual term is finite.
In order to work with infinite series and sequences, it is often necessary to know if they converge to a finite value.
This can be determined with a variety of techniques...
Divergence (nth-Term) Test
`\displaystyle\sum_{n=1}^\infty a_n \text{ diverges if } \lim_{n\to\infty}a_n\neq 0`
Direct Comparison Test
`\displaystyle (\forall n\in\mathbb N, 0 < a_n < b_n) \implies`
`\begin{cases}
\sum a_n \text{ converges} &\text{if }\sum b_n \text{ converges} \\
\sum b_n \text{ diverges} &\text{if }\sum a_n \text{ diverges}\end{cases}`
Alternating Series Test
`\displaystyle (\forall n\in\mathbb N, a_{n+1} \leq a_n \land \lim_{n\to\infty}a_n=0) \implies \sum_{n=1}^\infty (-1)^na_n \text{ converges}`
Integral Test
`\displaystyle (\forall x \in [1,\infty), f'(x) \leq 0 \land f(x) \geq 0) \implies \sum_{n=1}^\infty f(n) \text{ converges } \iff \int_1^\infty f(x) \,\mathrm d x \text{ converges}`
Ratio Test
`\displaystyle r=\lim_{n\to\infty}\lvert\frac{a_{n+1}}{a_n}\rvert \implies`
`\begin{cases}
\sum a_n \text{ converges} &\text{if } r < 1 \\
\text{Inconclusive} &\text{if } r = 1 \\
\sum a_n \text{ diverges} &\text{if } r > 1
\end{cases}`