Dobiński’s formula

Calculus (Integrals, Series)
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Riemann
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Dobiński’s formula

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Let $n \in \mathbb{N}$ and $\mathcal{B}_n$ denote the $n$ - th Bell number. Prove that

$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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