[Continuation of original comment]

We can look at it this way: We can try to assign values to a non-absolutely convergent series by bringing its terms in at less than full strength, producing an absolutely convergent series, and then increasing the terms' strengths towards full strength in the limit, observing what happens to the sum in the limit as well.

This is the idea behind the traditional account of series summation, mind you: at time T, we bring in all the terms of index < T at 100% strength and all other terms at 0% strength. This gives us our partial sums, and as T goes to infinity, each term's strength goes to 100%, so we can consider the partial sums as approximating the overall sum.

But we don't have to be so discrete as to only use 100% strength and 0% strength. We can try bringing in terms more gradually. For example, rather than having strengths discretely decay from 100% to 0% at some cut-off point, we can instead have the strengths decay exponentially in the index. (So at one moment, we may have the first term at 100% strength, the next term at 50% strength, the next term at 25% strength, etc.). Then we consider what happens as the rate of exponential decay slows, approaching no decay at all.

In symbols, this means we assign to a series a0 + a1 + a2... the limit, as b approaches 1 from below, of a0 * b^0 + a1 * b^1 + a2 * b^2 + .... Put another way, the limit, as h goes to 0 from above, of a0 * e^(-0h) + a1 * e^(-1h) + a2 * e^(-2h) + ..., where e is any fixed base you like. (Let's take e to be the base of the natural logarithm for convenience, and call this function of h the characteristic function of the series).

Again, this is not so different than the traditional account of series summation; we're just using exponential decay rather than sharp cutoff in our dampened approximations to the full series. (Actually, for the results we're interested in, it's really just the smoothness of the decay that's of interest. We could use other forms of smooth decay as well, and get the same results, but exponential decay is so convenient, I won't bother discussing in any further generality right now)

Now we've turned the question of determining the value of a series summation into the question of determining the limiting behavior of some function at 0.

Well, it's easy to determine limiting behavior at 0. Just write out a Taylor series centered at 0, and drop all the terms of positive degree, leaving only the term of degree 0. Boom, you've got the value of the function at 0.

Except... suppose the Taylor series has a few terms of negative degree as well. (As in, say, 5h^(-1) + 3 + 4h^2). Then the behavior at 0 isn't given by the degree 0 term; rather, the behavior at 0 is to blow up to infinity!

And, indeed, we'll find that this is precisely what happens when we look at the characteristic function of a series like 0 + 1 + 2 + 3 + ...; we get that f(h) = 0e^(-0h) + 1e^(-1h) + 2e^(-2h) + 3e^(-3h) + ... = e^(-h)/(1 - e^(-h))^2 = h^(-2) - 1/12 + h^2/240 - h^4/6048 + ....

Note that there is a negative degree term there. So in a very familiar sense, we can say that the behavior of this series is to blow up to infinity.

However, since any time a series DOES converge in the ordinary sense, the value it converges to is the degree 0 term of this characteristic function, it is very tempting and fruitful to think of the degree 0 term as the sum even when there are those pesky negative degree terms.

And in this more general sense, we see that the value of 0 + 1 + 2 + 3 + ... is that degree 0 term of f(h): -1/12. [In fact, we can understand the argument at the beginning of this post as outlining a rigorous calculation of this degree 0 term. (See https://www.quora.com/Mathematics/Theoretically-speaking-how... to see this spelt out)]

Now, you can propose other manipulations to produce other answers for this series in other ways, but this is one particular systematic account of summation which leads to this value alone and no other. [That is, for the series whose nth term is n. I should warn that, in the presence of negative degree terms in the characteristic function, this method is sensitive to index-shifting, so we would get different results if, for example, we considered 1, 2, 3, ... to be not the 1st, 2nd, 3rd, ..., terms, but rather the 0th, 1st, 2nd, ..., terms, respectively.]

Why should you care about this particular account of summation? Well, you don't have to; I can't force you to care about anything. But it's fairly natural and comes up with some significance in mathematics. It is, in a certain formal sense, precisely the account of summation which allows one to interpret the sum 1^n + 2^n + 3^n + ... for general complex n, yielding the Riemann zeta function (of great significance in number theory, and whose behavior (specifically, the Riemann hypothesis concerning its zeros) is generally considered one of the most important open problems in mathematics). So, you know, there's reason for some people to care about it, even if you don't.