Convergence
The sequence $\{a_n\}$ converges iff $\forall \varepsilon >0, \exists N\in\mathbb{N}\;\text{s.t.}\;n>N\implies |a_n-L|<\varepsilon$ $\iff \lim_{n\to\infty}a_n=L\text{ or }a_n\to L$
The Sandwich Theorem for Sequences $\displaystyle \forall n > N\; a_n\le b_n\le c_n,\, \lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L \implies \lim_{n\to\infty}b_n=L$
The Continuous Function Theorem for Sequences
bounded from above / upper bound / least upper bound bounded from below / lower bound / greatest lower bound
Geometric Series $\sum^\infty_{n=1} ar^{n-1} = \begin{cases} \frac{a}{1-r} & |r| < 1\\ \text{diverges} & |r| \ge 1 \end{cases}$
$n$th-Term Test $\sum^\infty_{n=1}a_n<\infty\implies a_n\to 0$
The Monotonic Sequence Theorem If a sequence $\{a_n\}$ is bounded and monotonic, then the sequence converges.
The Integral Test $\{a_n\}$에 대해 $a_n=f(n)$이라 하면 $f$가 $x\ge N$에 대해 continuous, positive, decreasing이면 $\sum^\infty_{n=N}a_n$ converge $\iff\int^\infty_Nf(x)\,dx$ converge
Bounds for the Remainder in the Integral Test $R_n=\sum a_i - \sum_1^n a_i \implies \int^\infty_{n+1}f(x)\,dx\le R_n\le \int^\infty_nf(x)\,dx$