A sequence is an ordered list of values indexed by the natural numbers. It can be viewed as a function

$$a:\mathbb N\to\mathbb R,$$

and its values are usually written

$$a_1,a_2,a_3,\ldots$$

or simply $(a_n)$.

Explicit and recursive descriptions

A sequence may be given by a direct formula, such as

$$a_n=\frac1n,$$

or recursively, where each term is defined from earlier terms. For example,

$$a_1=1,\qquad a_{n+1}=\frac12a_n$$

produces $1,1/2,1/4,1/8,\ldots$.

Convergence

A sequence converges to $L$ when its terms approach $L$ as the index grows:

$$\lim_{n\to\infty}a_n=L.$$

For example,

$$\lim_{n\to\infty}\frac1n=0.$$

If no finite limit exists, the sequence diverges.

Monotonicity and boundedness

A sequence is increasing when later terms never decrease and decreasing when later terms never increase. A sequence is bounded when all its terms remain within fixed upper and lower bounds.

Monotonicity and boundedness are useful because every bounded monotone real sequence converges.