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> since we don’t know Σ(n) we can’t know the exact value of Ω. Hmmm. Just because you don't know the individual terms of the sequence, it doesn't follow that y
by aaplok 4y ago
> since we don’t know Σ(n) we can’t know the exact value of Ω.
Hmmm. Just because you don't know the individual terms of the sequence, it doesn't follow that you don't know their infinite sum. Here's a trivial example: take the sum `(-1)^n Σ(floor(n/2))`. The infinite sum is obviously 0, yet we can't calculate each of the terms, nor many of the partial sums.
- alexmolas 4y ago> Just because you don't know the individual terms of the sequence, it doesn't follow that you don't know their infinite sum Yes, this is not true in general. But I think it's true for the specific definition of Ω we use in the post.
- aaplok 4y agoIntuitively I'd say you're probably right. But computation theory is full of counter-intuitive results. Your statement might be provable by adapting the argument that Σ(n) is not always computable. Something like computing the sum would require solving some version of the halting problem. Perhaps add a small edit to your article to highlight that this particular statement isn't to be taken as a mathematical proof?