sequences and series - Does $zeta(-1)=-1/12$ or $zeta(-1) to -1/12$?

I saw NumberPhile channel on Youtube, and they proved $1+2+3+\cdots=-1/12$. Also, I read This.

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So, which one is correct

$$\zeta(-1)=-1/12\\ \text{or} \\\zeta(-1) \to -1/12$$

Equivalent to:

$$1+2+3+\cdots=-1/12\\ \text{or} \\1+2+3+\cdots \to -1/12$$

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My question: Does it "equal" or "converge"?

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Question Explanation:

I mean by "$\to$" "approaches to", like $x\to a$ means $\forall \epsilon>0, |x-a|<\epsilon.$