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.


  • 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.


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.


  • 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 +∞.

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