# Series

## Series

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 +∞.