/sci/ - Science & Math

>N is bounded from above
>no j/k it's bounded from below outside of N

legit mad

>>6340416
Which bit is that?

>>6340404
its not a "real" sum, its a number that acts like a sum for divergent series, such as as you add two series term by term, then the sums also get added, its better to think of it as a function that takes an infinite number of variables, and return a finite number, f(X) = f(x1, x2, x3, ...) = y, with the attributes such that f(a*x1, a*x2, a*x3, ...) = a*y, and f(X+Y) = f(X) + f(Y), and such that if the sum x1 + x2 + x3 + ... converges to a limit then f(x1, x2, x3, ...) is equal to the limit. this last property is what makes it act so close to a sum.

another way to think about it which shows the physical relevance is that you separate the infinity from the rest of the answer, and in a lot of physical systems, such as when you are working with energy is QFT or ST where you use this sometimes, you dont care about the actual energy, but only differences in energy, so as long as you subtract the same infinity from all energies, it will work. in the case of 1 + 2 + 3 + 4 + .. you can manipulate the series to get 1/ep^2 - 1/12 + ep + ep^2/ + ... with the limit of ep going to 0, thus you get infinity - 1/12, but we dont care about the infinity so we dont write it and only write -1/12.

if you can find a proof that the Riemann Zeta function is valid for -1, is anything else missing? For a proof to be correct, it also has to be complete. until you find a validity proof, theres a hole in that particular proof.

>>6340444
you cant really prove it, since its a definition. all you can prove is that using analytical continuation for a divergent sum adheres to the conditions of summation methods; as in >>6340438

>>6340460
And can that be proved in this case? And if so, does that prove the result of -1/12 to be correct?

>>6340470
yes, ill type it a bit later if no one else provides anything in the meantime.

This shit is only the answer when dealing with the set of reals with complex numbers right? Not just the set of integers? How could you get a non integer from the sum of all integers?

So its correct in a certain regard but that doesn't mean intuition and common sense about the sum of all integers is wrong.

>>6340481
Cool, thanks.

>>6340483
I doubt there is any natural intuition about infinite sums, thats exactly why this result is so unintuitive

>>6340481
Are you still here anon?

Found an article some time ago which provided a simple counterexplanation, don't have it on this computer too, try looking it up.

>>6340643
actually i was wrong, you cant prove its stable and linear for the case 1+2+3+... you can only prove that any method is ether stable or linear, and not both. thus you have to go to the weaker definition without stability that uses finite reindexability.