section_path: Uses/Series
section_name: Series
paragraph_text: Series
  
  A particular expression of interest which is formalized as the limit of a sequence is sums of infinite series. These are "infinite sums" of real numbers, generally written as ∑ n = 1 ∞ a n. {\displaystyle \sum _{n=1}^{\infty }a_{n}.} This is defined through limits as follows: given a sequence of real numbers { a n } {\displaystyle \{a_{n}\}}, the sequence of partial sums is defined by s n = ∑ i = 1 n a i. {\displaystyle s_{n}=\sum _{i=1}^{n}a_{i}.} If the limit of the sequence { s n } {\displaystyle \{s_{n}\}} exists, the value of the expression ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} is defined to be the limit. Otherwise, the series is said to be divergent.
  
  A classic example is the Basel problem, where a n = 1 / n 2 {\displaystyle a_{n}=1/n^{2}}. Then ∑ n = 1 ∞ 1 n 2 = π 2 6. {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}.}
  
  However, while for sequences there is essentially a unique notion of convergence, for series there are different notions of convergence. This is due to the fact that the expression ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} does not discriminate between different orderings of the sequence { a n } {\displaystyle \{a_{n}\}}, while the convergence properties of the sequence of partial sums can depend on the ordering of the sequence.
  
  A series which converges for all orderings is called unconditionally convergent. It can be proven to be equivalent to absolute convergence. This is defined as follows. A series is absolutely convergent if ∑ n = 1 ∞ | a n | {\displaystyle \sum _{n=1}^{\infty }|a_{n}|} is well defined. Furthermore, all possible orderings give the same value.
  
  Otherwise, the series is conditionally convergent. A surprising result for conditionally convergent series is the Riemann series theorem: depending on the ordering, the partial sums can be made to converge to any real number, as well as ± ∞ {\displaystyle \pm \infty }.