# Summary: Series and Their Notations

## Key Equations

sum of the first [latex]n[/latex] terms of an arithmetic series | [latex]{S}_{n}=\frac{n\left({a}_{1}+{a}_{n}\right)}{2}[/latex] |

sum of the first [latex]n[/latex] terms of a geometric series | [latex]{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\cdot r\ne 1[/latex] |

sum of an infinite geometric series with [latex]-1<r<\text{ }1[/latex] | [latex]{S}_{n}=\frac{{a}_{1}}{1-r}\cdot r\ne 1[/latex] |

## Key Concepts

- The sum of the terms in a sequence is called a series.
- A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
- The sum of the terms in an arithmetic sequence is called an arithmetic series.
- The sum of the first [latex]n[/latex] terms of an arithmetic series can be found using a formula.
- The sum of the terms in a geometric sequence is called a geometric series.
- The sum of the first [latex]n[/latex] terms of a geometric series can be found using a formula.
- The sum of an infinite series exists if the series is geometric with [latex]-1<r<1[/latex].
- If the sum of an infinite series exists, it can be found using a formula.
- An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.

## Glossary

**annuity**an investment in which the purchaser makes a sequence of periodic, equal payments

**arithmetic series**the sum of the terms in an arithmetic sequence

**diverge**a series is said to diverge if the sum is not a real number

**geometric series**the sum of the terms in a geometric sequence

**index of summation**in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation

**infinite series**the sum of the terms in an infinite sequence

**lower limit of summation**the number used in the explicit formula to find the first term in a series

**nth partial sum**the sum of the first [latex]n[/latex] terms of a sequence

**series**the sum of the terms in a sequence

**summation notation**a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series

**upper limit of summation**the number used in the explicit formula to find the last term in a series

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].

### CC licensed content, Specific attribution

- Precalculus.
**Provided by:**OpenStax**Authored by:**OpenStax College.**Located at:**https://cnx.org/contents/[email protected]:1/Preface.**License:**CC BY: Attribution.