$${a_n}^\infty_1$$

$$s_n = \sum_{i=1}^{n} a_i$$

$$s_\infty = \lim\limits_{n \to \infty} \sum_{i=1}^{n} a_i$$

A list of terms related by a common pattern.

A sequence where $\lim_{n \to \infty} a_n$ exists and is finite.

A sequence where $\lim_{n \to \infty} a_n$ does not exist or is infinite.

A sequence where $\frac{a_{n+1}}{a_n} > 1$ for all n.

A sequence where $\frac{a_{n+1}}{a_n} < 1$ for all n.

A sequence that is either increasing or decreasing.

There exists an upper bound to the sequence.

There exists a lower bound to the sequence.

A sequence that is both bounded above and below.

A sum of the terms in a sequence.

The value of the summation of the 1st through the $n^{th}$ terms.

A series where n=$infty$, or $s_\infty = \lim\limits_{n \to \infty} \sum_{i=1}^{n} a_i$.

A series in which $s_\infty$ exists and is finite.

A series in which $s_\infty$ does not exist or is infinite.

A series where the middle terms cancel out, leaving only the first and last terms.

A sequence converges if its limit as n approaches infinity exists and is a finite number. It approaches a specific value.

A sequence diverges if its limit as n approaches infinity does not exist (oscillates) or is infinite. It does not approach a specific value.

A sequence is a list of numbers, while a series is the sum of the numbers in a sequence.

The sum of the infinite terms approaches a finite value.

The sum of the infinite terms does not approach a finite value; it either goes to infinity or oscillates.