 If the latter series converges for 0 < "x" < 1 to a function with a limit as "x" tends to 1, then this limit is called the Abel sum of the original series, after Abel's theorem which guarantees that the procedure is consistent with ordinary summation.For Grandi's series one has

The corresponding calculation that the Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + · · · is 2⁄3 involves the function (1 + "x")/(1 + "x" + "x"2).Whenever a series is Cesàro summable, it is also Abel summable and has the same sum.