# sequence

A sequence, also called a **progression**, is an ordered set of numbers (e.g.,
2, 4, 6, 8, ...2*n*, ...), which may be finite or infinite in length.
The members of the sequence are linked by a common mathematical formula.
Infinite sequences may or may not be convergent (tend to a single limit).
The sequence 0, 0.9, 0.99, 0.999, ... for example is convergent, tending
to the limit 1. Among the different types of sequence are arithmetic
sequences, geometric sequences,
and harmonic sequences.

A **unmilled sequence** is a sequence that first increases and then decreases.

## Point-sequence

If according to some prescribed rule a first point *P*_{1} determines a second *P*_{2}, and *P*_{2} determines a third *P*_{3}, and so on, these points are said to form a point-sequence:

*P*_{1}, *P*_{2}, *P*_{3}, *P*_{4}, ...

The points dealt with can be either in a plane or in three-dimensional space. Point-sequences are closed later to number-sequences. Thus, if the points are specified by their coordinates, three number sequences are formed from the coordinates *P _{n}* (

*x*,

_{n }*y*,

_{n}*z*) of the general point

_{n}

*x*_{1}, *x*_{2}, *x*_{3}, ..., *x _{n}*, ...

*y*_{1}, *y*_{2}, *y*_{3}, ..., *y _{n}*, ...

*z*_{1}, *z*_{2}, *z*_{3}, ..., *z _{n}*, ...

Conversely, given an arbitrary number-sequence *a*_{1}, *a*_{2}, *a*_{3}, ..., a point-sequence can be derived in which the number *a _{n}* is associated with the point

*A*(

_{n }*a*, 0) of the plane (

_{n}*x*,

*y*).

A point-sequence is said to *converge* to the point *A* (*P _{n} → A*) if the distance

*P*form a null sequence. A point-sequence which is not convergent is said to be

_{n}A*divergent*. E.g. suppose the following points are given in a plane:

The sequence converges to the point *P *(2, 4), since the sequence of distances

is a null sequence.