The Abel-regularized sum, also called the Abel sum, of a series 
is the limit
 |
(1)
|
when the power series converges for 
and the displayed limit
exists (Hardy 1949, p. 7). If the original series
converges in the ordinary sense, Abel's convergence
theorem guarantees that its Abel-regularized sum equals its usual sum.
The converse need not hold, so Abel regularization assigns values to some divergent
series.
Writing 
for the partial sums gives the equivalent expression
 |
(2)
|
for 
.
Thus, Abel regularization uses geometrically weighted partial
sums. For example, 
has Abel-regularized sum 
. For the series 
,
 |
(3)
|
for 
,
so its Abel-regularized sum is 
.
The power series convergence requirement excludes 
,
whose radius of convergence in 
is 
. Evaluating the analytic
continuation 
at 
gives 
,
but does not give an Abel-regularized sum of that series.
Every series having a Cesàro sum of any positive integer order has the same Abel-regularized sum (Hardy 1949, p. 108).
