All such attempts for summation are ill-defined. The Abel summation is actually not a property - it is a functional. That said you take the function f(n)=<n-th

All such attempts for summation are ill-defined. The Abel summation is actually not a property - it is a functional. That said you take the function f(n)=<n-th summand>. Then you can define AS(f), but it shares very few properties with normal summation. If you are not careful you can easily reach contradiction. For example 1/4 = 1 -2 +3 -4 +5 -6 +7 ... = //comment: 1-2-4+5=0, 3-6+9-12... = 3\sum i(-1)^i= 3* 1/4 Contradiction! Edit: The original contradiction example was more complicated.