Series
Real number that can be expressed as the sum of an infinite sequence of real numbers, that is, an expression of the form [latex]a_1 + a_2 + a_3 + ... + a_n[/latex], or, in abbreviated form, [latex]\sum\limits_{i=1}^{+infty }a_i[/latex] where [latex]a_i[/latex] are the terms of an infinite numerical sequence {[latex]a_n[/latex]} 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 [latex]\sum\limits_{i=1}^{+infty }a_i[/latex] is read as "the sum of all the terms [latex]a_i[/latex], where i takes the values from 1 to +∞". The notation [latex]S_n[/latex] is used to write the sum of a finite sequence of the first n terms of a sequence.Examples
- The series [latex]S_n[/latex] = 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 [latex]\dfrac{3}{10} + \dfrac{3}{100} + \dfrac{3}{1000} + \dfrac{3}{10 000} + ... [/latex] is a convergent series whose limit is the rational number [latex]\dfrac{1}{3}[/latex] or the repeating decimal [latex]0.\overline{3}[/latex].
- The series [latex]S_n = 1 + 2 + 3 + 4 + ... + n = \dfrac{n(n+1)}{2}[/latex] is divergent. Its limit is +∞.