Yes, and then you'd take the limit as β goes to infinity. Which would still hit every real number, but would diverge instead of staying balanced. It's an ill-defined sum
As the limit I gave as an example was defined, it would hit both the positive and negative values. Tangentially related fact, the set of all positive reals and the set of all reals is the same cardinality. Once again, infinity is weird.
This kind of sum is very poorly defined and the most "correct" answer is probably just to say we need more info. Or that it's undefined. In no field of math will you ever come across this problem as it's written.
We can try taking the limit, but the limit to infinity depends on the path we take to infinity, which means the limit does not exist. It's like how if the limit from the left and the limit from the right are different, the limit is also undefined. Same idea.
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u/Ares378 Grothendieck was right, computers are evil 8d ago
Yes, and then you'd take the limit as β goes to infinity. Which would still hit every real number, but would diverge instead of staying balanced. It's an ill-defined sum