"Comparing coefficients" in the context of Bernoulli numbers

Question

I'm currently studying Bernoulli numbers and I ended up with $$1=\displaystyle\sum_{k=0}^\infty \left(\displaystyle\sum_{n=0}^k \dbinom{k+1}{n}\mathrm B_{n}\right)\dfrac{z^k}{(k+1)!}.$$ Now, "comparing coefficients" should yield $$\displaystyle\sum_{n=0}^k \dbinom{k+1}{n}\mathrm B_{n}=0,$$ but I don't understand that step of "comparing coefficients". Also, why are the Bernoulli numbers extracted from this correct only for $k\gt 0$? What caused the loss of generality?