 Bernoulli formula for zeta values at nonpositive integers

For every integer $n\ge 1$, the Riemann zeta function satisfies $\zeta (1-n)=(-1{)}^{n-1}{B}_n/n$. One proof applies meromorphic continuation of a Mellin transform from an asymptotic expansion to $(e^y-1{)}^{-1}$, whose transform is the Bose integral $\Gamma (s)\zeta (s)$, and divides residues by the residues of the Gamma function. The formula includes $\zeta (0)=-1/2$.