Why does $1+2+3+4+\dots=-\frac1{12}$ in a couple different ways? $1+2+3+4+\dots$ is undefined when using regular summation. If you use either Ramanujan summation or Zeta function regularization, then $1+2+3+4+\dots=-\frac1{12}$. This article lists some over definitions of summation, and they all give the same result.
My question is, why do all these seemingly unrelated definitions all give the same, seemingly arbitrary value of $-\frac1{12}$ to these seemingly simple divergent series? Is there only underlying method that they all are based on?

First note the Abel summation gives $\infty$.
Now I'd say your observed phenomenon is mainly because $(1-2^{1-s}) \zeta(s) = \eta(s)$ where $\eta(s)= \sum_{n=1}^\infty (-1)^{n+1} n^{-s}$ can be extended analytically to $\Re(s) > -K$ just by partial summation (since $\sum_{n=1}^N (-1)^{n+1} = \frac{1+(-1)^{N+1}}{2}$) and hence also by Abel summation.
Therefore $$\zeta(-1)(1-2^{2}) = \eta(-1) = \lim_{z \to 1^-} \sum_{n=1}^\infty (-1)^{n+1}n z^n=\lim_{z \to 1^-} z\frac{d}{dz}\frac{-z}{1+z} =\frac{1}{4}$$
and the same will happens for any regularization method compatible with the Abel summation on rational functions.
