Is there a good shift in perspective to understand the relation between $frac{-1}{12}$ and $1+2+3+4+5+6+7+…$. I do understand how to see it using the Riemann Zeta Function. I heard how Ramanujan found it, And I found this question on it Is there a metric in which 1+2+3+4+… converges to -1/12?. I understand how in some way that it’s equal.

But I understand why in some way $1+2+4+8+16+…=-1$ better because even though it’s not getting closser to -1 in the normal sense of distance, but if you change your definition of distance with some of the same nice properties mainly the 2-adic metric. I found this explaination from this video -> https://www.youtube.com/watch?v=XFDM1ip5HdU

Is their a similar way to move a new perspective a new metric or something to see that in a way $1+2+3+4+5+6+7+…=frac{-1}{12}$. If it is does it make that work will it work with $1+4+9+16+25+36+49+…=0$? Thanks in adivance I’ve been thinking about it for days.