# Infinite Sequences

## Key Questions

• An infinite sequence of numbers is an ordered list of numbers with an infinite number of numbers.

An infinite series can be thought of as the sum of an infinite sequence.

• The sequence $\left\{{a}_{n}\right\}$ converges if ${\lim}_{n \to \infty} {a}_{n}$ exists (having a finite value); otherwise, it diverges.

I hope that this was helpful.

• Informally, and (real-valued) infinite sequence is just an infinite list of real numbers ${x}_{1} , {x}_{2} , {x}_{3} , {x}_{4} , \setminus \ldots$.

More precisely, an infinite sequence is a function whose domain can be taken (among other things) to be the set of positive integers $\mathbb{N} = \setminus \left\{\setminus 1 , 2 , 3 , 4 , \setminus \ldots \setminus\right\}$ and whose codomain is the set of real numbers $\mathbb{R}$. The output of the sequence at the input $n \setminus \in \mathbb{N}$ is ${x}_{n} \setminus \in \mathbb{R}$.