## Mathematical Analysis## Sequences and Series |

## What is a series? |

A series is closely related to the sum of numbers. It is actually used to help add numbers; therefore, in a sequence it can be the indicated sum of that sequence. In general, the idea is to start with a number, then do something to that number to get the next number, then do the same to that number to get the next number, and so on. For example, a *finite series* with six terms is 2 + 4 + 6 + 8 + 10 + 12, in which 2 is added to each number to get the next number. An example of an *infinite series* is one with the notation 1/2^{n}, with *n* ≥ 1, or ½ + 1/4 + 1/8 + … (and so on).

To see a series written in notation, if set {*x*_{n}} is a sequence of numbers being added, and set *s*_{1} = *x*_{1}, then *s*_{2} = *x*_{1} + *x*_{2}; *s*_{3} = *x*_{1} + *x*_{2} + *x*_{3}; and so on. And for *n* ≥ 1, a new sequence is made, {*s*_{n}}, called the *sequence of partial sums*, or *s _{n}* =

*x*

_{1}+

*x*

_{2}+ … +

*x*

_{n}