section_path: Generalizations/Summations over arbitrary index sets/Well-ordered sums
section_name: Well-ordered sums
paragraph_text: Well-ordered sums
  
  Conditionally convergent series can be considered if I {\displaystyle I} is a well-ordered set, for example, an ordinal number α 0. {\displaystyle \alpha _{0}.} In this case, define by transfinite recursion:
  
  ∑ β < α + 1 a β = a α + ∑ β < α a β {\displaystyle \sum _{\beta <\alpha +1}a_{\beta }=a_{\alpha }+\sum _{\beta <\alpha }a_{\beta }}
  
  and for a limit ordinal α, {\displaystyle \alpha,}
  
  ∑ β < α a β = lim γ → α ∑ β < γ a β {\displaystyle \sum _{\beta <\alpha }a_{\beta }=\lim _{\gamma \to \alpha }\sum _{\beta <\gamma }a_{\beta }}
  
  if this limit exists. If all limits exist up to α 0, {\displaystyle \alpha _{0},} then the series converges.