## Mathematical Analysis## Sequences and Series |

## What is a sequence? |

A sequence is defined as a set of real numbers with a natural order. A sequence is usually included in brackets ({}), with the *terms*, or parts of a sequence, separated by commas. For example, if a scientist collects weather data every day for many days, the first day of collecting can be written as *x _{1}* data; then

*x*for the second day, and so on until

_{2}*x*

_{n}, in which

*n*is the eventual number of days. This can be written as {

*x*

_{1},

*x*

_{2}, …

*x*

_{n}}

_{n≥1}. In general, the sequence of numbers in which

*x*is the

_{n}*nth*number is written using the following notation: {

*x*

_{n}}

_{n≥1}.

A sequence can get larger or smaller. For example, in the sequence for {2^{n}}_{n≥1}, the solution is 2 ≤ 4 ≤ 8 ≤ 16 ≤ 32, and so on, with the numbers getting larger. Whereas, for {1/*n*}_{n≥1}, the sequence becomes 1 ≥ ½ ≥ 1/3 ≥ 1/4 ≥ 1/5, and so on, with the numbers getting progressively smaller. This does not mean that sequences only get progressively larger and smaller; certain solutions for sequences include a mix of the two.