## Mathematical Analysis## Sequences and Series |

## When is a sequence monotonic? |

A sequence is called monotonic if one of the following properties hold: In the sequence {*x*_{n}}_{n≥1}, it is increasing if and only if *x _{n} <>_{n} +*

_{1}for any

*n*≥ 1, or it is decreasing if and only if

*x*

_{n}> x_{n}+_{1}for any

*n*≥ 1.

For example, in order to check that the sequence {2^{n}}_{n≥1} is increasing: Let *n* ≥ 1; that gives 2^{n+1} = 2^{n} 2. Because 2 is greater than 1, which means that 1 × 2^{n} < 2="" ×="">^{n}; thus 2^{n} <>^{n + 1}, which shows the sequence is increasing.