section_path: Generalizations/Divergent series
section_name: Divergent series
paragraph_text: Divergent series
  
  Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, (C, k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).
  
  A variety of general results concerning possible summability methods are known. The Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.