The sum of the natural numbers is equal to infinity (in the standard Euclidean metric on the extended real line using a constant, discrete unbounded measure). We could change those constraints. For example, if we wanted the sum to actually be -1/12, we could take the one-point compactification of the real number line (which is a circle, mind you), then identify the "point at infinity" with -1/12 to get a quotient topology on the reals. In this topology, the sum actually converges to -1/12. We could also be sillier and use, say, the finite complement topology or the indiscrete topology. Both of those are non-Hausdorff, which means limits may not be unique, and indeed in both of those topologies, the sum of all natural numbers is everything. Literally, any number you could tell me would be the right answer. On the other hand, we could leave the Euclidean metric alone and mess with the measure. If we take Lebesgue measure (the measure we usually work with when taking integrals), we would turn the sum into an integral over the natural numbers. The set of natural numbers has measure zero, so the sum (integral) in this case is zero. We could take a finite measure where each natural number is weighted differently, but then all you're doing is finding the total probability of an event, and at that point you're just doing statistics. And if we're doing probability, the probability of a 0 outcome may be nonzero, so we need to agree on whether or not 0 is a natural number (it is; it categorically is; some wrong people like to disagree with me on that). So, I argue the answer to your question "what is the sum of the natural numbers" is: "define every word in that sentence".

And that's what mathematics is, kids. Changing the rules like a toddler until you get the answer you want