Denote by $B_n$ the Bernoulli sequence (defined by the exponential generating function $\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$), and let $c_n$ be a sequence of real (or complex) numbers such that the series
$\sum^{\infty}_{n=0}\frac{n!}{\pi^{n}}c_n$ is absolutely convergent. Then we would like to prove that
$$ \lim_{n\rightarrow \infty}\frac{1}{n!}\sum^{\infty}_{j=0}(-1)^j\frac{(j+n-1)!}{j!}B_jc_{j+n-1}=0. $$