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MacLaurin of exp(exp(x))

Let \mathcal{B}_n denote the n-th Bell number. Prove that

    \[\exp \left(\exp(x) \right) = e \sum_{n=0}^{\infty} \frac{\mathcal{B}_n}{n!} x^n\]

Solution

Taking derivatives we get that

    \[f^{(n)}(x)=\sum_{k=0}^n{n\brace k}\exp(e^x+kx)\]

where f(x)=\exp \left(\exp(x) \right) and n\brace k are the Stirling numbers of second kind. We also note that

    \[f^{(n)}(0)=e\sum_{k=0}^n{n\brace k}=e\mathcal{B}_n\]

where \mathcal{B}_n are the Bell numbers. Thus,

    \[exp(\exp(x))=e\sum_{n=0}^\infty\frac{\mathcal{B}_n}{n!}x^n\]

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