Real number that can be expressed as the sum of an infinite sequence of real numbers, that is, an expression of the form \(a_1 + a_2 + a_3 + … + a_n\), or, in abbreviated form, \(\sum\limits_{i=1}^{+infty }a_i\) where \(a_i\) are the terms of an infinite numerical sequence {\(a_n\)} of real numbers.

A series is a sum of terms, whereas a numerical sequence is a list of terms. In both cases, the order of the terms follows an arithmetic rule.

### Property

- A series is called
**finite**or**infinite**depending on whether it contains a finite or an infinite number of terms. The term*infinite series*is a synonym for*series*. - An
**arithmetic series**is a series whose terms are elements of an arithmetic sequence. - A
**convergent series**is a series that approaches a given number. - A
**divergent series**is a series that is not convergent.

### Symbol

The Greek capital letter *sigma*, “Σ”, is used to indicate a summation in abbreviated form.

The expression \(\sum\limits_{i=1}^{+infty }a_i\) is read as “the sum of all the terms \(a_i\), where *i* takes the values from 1 to +∞”.

The notation \(S_n\) is used to write the sum of a finite sequence of the first *n* terms of a sequence.

### Examples

- The series \(S_n\) = 0 + 1 + 2 + 3 + 4 + … +
*n*is an arithmetic series that is the sum of the consecutive whole numbers up to*n*. - The series of decimal fractions \(\dfrac{3}{10} + \dfrac{3}{100} + \dfrac{3}{1000} + \dfrac{3}{10 000} + … \) is a convergent series whose limit is the rational number \(\dfrac{1}{3}\) or the repeating decimal \(0.\overline{3}\).
- The series \(S_n = 1 + 2 + 3 + 4 + … + n = \dfrac{n(n+1)}{2}\) is divergent. Its limit is +∞.