section_path: Basic properties
section_name: Basic properties
paragraph_text: Basic properties
  
  An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form
  
  a 0 + a 1 + a 2 + ⋯, {\displaystyle a_{0}+a_{1}+a_{2}+\cdots,}
  
  where (a n) {\displaystyle (a_{n})} is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (for instance elements of any abelian group in abstract algebra). This is an expression that is obtained from the list of terms a 0, a 1, … {\displaystyle a_{0},a_{1},\dots } by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
  
  ∑ n = 0 ∞ a n. {\displaystyle \sum _{n=0}^{\infty }a_{n}.}
  
  If an abelian group A of terms has a concept of limit (e.g., if it is a metric space), then some series, the convergent series, can be interpreted as having a value in A, called the sum of the series. This includes the common cases from calculus, in which the group is the field of real numbers or the field of complex numbers. Given a series s = ∑ n = 0 ∞ a n {\textstyle s=\sum _{n=0}^{\infty }a_{n}}, its k th partial sum is
  
  s k = ∑ n = 0 k a n = a 0 + a 1 + ⋯ + a k. {\displaystyle s_{k}=\sum _{n=0}^{k}a_{n}=a_{0}+a_{1}+\cdots +a_{k}.}
  
  By definition, the series ∑ n = 0 ∞ a n {\textstyle \sum _{n=0}^{\infty }a_{n}} converges to the limit L (or simply sums to L), if the sequence of its partial sums has a limit L. In this case, one usually writes
  
  L = ∑ n = 0 ∞ a n. {\displaystyle L=\sum _{n=0}^{\infty }a_{n}.}
  
  A series is said to be convergent if it converges to some limit, or divergent when it does not. The value of this limit, if it exists, is then the value of the series.