The Abel-regularized sum of a series 
is the limit
 |
when the power series converges for 
and the displayed limit exists. If the original
series converges in the ordinary sense, its Abel-regularized
sum equals its usual sum. The converse need not hold, so
Abel regularization assigns values to some divergent
series.
For example, 
has Abel-regularized sum 
. Every series
having a Cesàro sum is Abel-summable to the
same value.
