sequences and series - Value of Riemann zeta function at $-1$

This claim is false $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}n^{-(-1)}= \zeta(-1)=-1/12$.

The error is that we should

$\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}(1/n ^1)^{-1}=(0)^{-1}$.

Am I correct? It's difficult to say that an infinite sum like that don't diverge and that sum of positive numbers can give negative number.