convergence of an iterated series which is had Bernoulli numbers

Question

As we know the Bernoulli numbers $B_n$ are a sequence of signed rational numbers that can be defined by the exponential generating function $$\frac{x}{e^x-1}=\sum^{\infty}_{n=0}\frac{B_n x^n}{n!}.$$ Now my question is here. Is this series convergent, for all $z\in \mathbb{C}$? $$\sum^{\infty}_{t=1}\sum_{k=0}^{\infty}\frac{(-1)^{k+t+1}}{(t+1)!} ‎\frac{B_{2k}}{(2k)!(t+2k+1)}‎z^{t+1}$$