By using Cesàro summation you can get:
Mathologer at 18:28:
This video explains a lot of things dealing with infinite sums.
sum can be 0, 1 or 1/2
depending on the method by which you tyr to solve the question.
although it is upto infinity so nothing could be said exactly.
There is only ONE method to assign a finite value (if it exits) to an infinite sum and that method is really a definition.
If the infinite sequence of partial sums converges to a finite value then we assign that finite value to the infinite sum. If the infinite sequence of partial sums diverges then the infinite series diverges. No other method is correct or even makes any sense.
s1 = 1
s2 = 0
s3 = 1
s4 = 0
Clearly this sequence does not converge. Therefore the original sum does not converge to any finite value and is therefore divergent.
You are right. It is exactly what the Wikipedia article and Mathologer explain.
Anyway, there is something noticeable with Grandi's partial sums. They converges on average to 1/2. It is just another way to apprehend it as far as it satisfies some well defined conditions.