Definition: An ordered list of numbers $\{a_n\}_{n=1}^\infty$.
Convergence: A sequence $\{a_n\}$ converges to $L$ if $\lim_{n \to \infty} a_n = L$. Otherwise, it diverges.
Properties: Sum, difference, product, quotient, constant multiple rules apply to limits.
Monotonic Sequence: Either non-decreasing ($a_n \le a_{n+1}$) or non-increasing ($a_n \ge a_{n+1}$).
Bounded Sequence: Bounded above ($a_n \le M$) and bounded below ($a_n \ge m$).
Monotonic Sequence Theorem: If a sequence is both monotonic and bounded, it converges.

Definition: Sum of an infinite sequence $S = \sum_{n=1}^\infty a_n$.
Partial Sums: $S_N = \sum_{n=1}^N a_n$.
Convergence: The series converges if the sequence of partial sums $\{S_N\}$ converges to a finite limit $S$. Otherwise, it diverges.
Geometric Series: $\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$ if $|r| < 1$. Diverges if $|r| \ge 1$.
$p$-series: $\sum_{n=1}^\infty \frac{1}{n^p}$. Converges if $p > 1$, diverges if $p \le 1$.
Test for Divergence (nth Term Test): If $\lim_{n \to \infty} a_n \ne 0$, then $\sum a_n$ diverges. (If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive).

If $f(x)$ is positive, continuous, and decreasing for $x \ge 1$, then $\sum_{n=1}^\infty a_n$ and $\int_1^\infty f(x) dx$ either both converge or both diverge. (where $a_n = f(n)$).

Direct Comparison Test: Assume $0 \le a_n \le b_n$ for all $n$.
- If $\sum b_n$ converges, then $\sum a_n$ converges.
- If $\sum a_n$ diverges, then $\sum b_n$ diverges.
Limit Comparison Test: Assume $a_n > 0$ and $b_n > 0$. If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Absolute Convergence: A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges. If a series converges absolutely, then it converges.
Conditional Convergence: A series converges but does not converge absolutely.
Ratio Test: Let $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.
- If $L < 1$, the series converges absolutely.
- If $L > 1$ or $L = \infty$, the series diverges.
- If $L = 1$, the test is inconclusive.
Root Test: Let $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$.
- If $L < 1$, the series converges absolutely.
- If $L > 1$ or $L = \infty$, the series diverges.
- If $L = 1$, the test is inconclusive.

Alternating Series: Series of the form $\sum (-1)^{n-1} b_n$ or $\sum (-1)^n b_n$, where $b_n > 0$.
Alternating Series Test (Leibniz): If an alternating series satisfies:
1. $b_n$ is decreasing ($b_{n+1} \le b_n$)
2. $\lim_{n \to \infty} b_n = 0$
Then the series converges.
Estimation: For a convergent alternating series, $|R_N| = |S - S_N| \le b_{N+1}$.