Is there a metric in which 1+2+3+4+... converges to -1/12?
Solution 1:
Here is an example (though it's a bit silly). Let $f : \mathbb{R} \to \mathbb{R}$ be some bijection such that $f(n) = \frac{1}{n}$ and $f(\frac{-1}{12}) = 0$, and define a metric on $\mathbb{R}$ by $d(x,y) = |f(x) - f(y)|$. You can easily check that this satisfies all the axioms of a metric. And $d(1+2+\cdots+n, -1/12) = 1/(1+2+\cdots+n) \to 0$, so for this metric the series $\sum_k k$ converges to $-1/12$.
But as you can see the metric doesn't have a lot to do with the usual metric on $\mathbb{R}$, or with the usual addition/multiplication.