Another summation method is Abel summation. The idea is, if we have
then we can define
for . Then if exists and is finite, that limit is the Abel sum of .
As a simple example, apply this to Grandi’s series, . Here for . The limit as is , the same result as we obtained using Cesàro summation. In fact it can be shown that Abel summation is stronger than Cesàro summation, i.e., for series that can be Cesàro summed, Abel summation give the same result, but there are additional series which can be Abel summed but not Cesàro summed. Of course Cesàro summation is consistent with ordinary summation for convergent series, and therefore so is Abel summation: that is, Abel summation is regular.
Here’s another example. Consider
Not only does this series not converge, but the partial sum averages don’t converge either, so it is not Cesàro summable. But it is Abel summable. We make this sum into a function
But now notice: , , , and so on:
and, again, for ,
Now the Abel sum is .
A couple more properties (besides regularity) a summation method might have are linearity and stability. For the following let denote the result of applying summation method to series . By linearity is meant: if and then . By stability is meant: if then , and conversely. Cesàro summation and Abel summation both are linear and stable. So is classical summation, for that matter.
You can prove that for any linear and stable summation method , the sum of the Grandi series is , if that sum exists:
That “if that sum exists” provision is important. For instance, classical summation of the Grandi series is undefined, not , even though classical summation is linear and stable. You can come up with similar proofs about linear and stable sums of other series, that they must always have some particular value if they have a value at all. Showing that they do indeed have a value is another matter!
Conversely, you can prove some series do not have values for any summation method that is linear and/or stable. For example, suppose is stable and has value . Then
an impossibility. So cannot be summed by any stable summation method. There are unstable methods, however, that can sum that series.