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A series where most terms cancel out when summed. Explicit formula varies, but typically has a form where $S = \sum (b_n - b_{n+1})$ results in many terms canceling out.

A type of power series that represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Formula: $f(x) = f(c) + f'(c)(x - c) + \frac{f''(c)}{2!}(x - c)^2 + ... + \frac{f^{(n)}(c)}{n!}(x - c)^n + ...$

A geometric series with infinite terms. Formula for the sum: $S = \frac{a_1}{1 - r}$, where $|r| < 1$

A sequence in which each term after the first is obtained by adding a constant difference. Formula: $a_n = a_1 + (n-1)d$

A sequence of numbers formed by taking the reciprocals of an arithmetic sequence. No specific formula, but can be expressed as $a_n = \frac{1}{a_1 + (n-1)d}$ where $d$ is the difference between the terms in the underlying arithmetic sequence.

A sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1. Formula: $F_n = F_{n-1} + F_{n-2}$

An infinite series of the form $\sum_{n=0}^\infty a_n(x - c)^n$ where $a_n$ represents the coefficients and $c$ is a constant.

A series of the form $\sum \frac{1}{n^p}$, where $p$ is a positive constant. Converges when $p > 1$ and diverges when $p \leq 1$.

A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Formula: $a_n = a_1 \cdot r^{(n-1)}$

The summation of a geometric sequence. Formula for the sum of the first $n$ terms: $S_n = a_1\frac{1 - r^n}{1 - r}$, where $r \neq 1$

The summation of an arithmetic sequence. Formula for the sum of the first $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n)$