infinite product of n=0 to infinity of (1+1/e^n)^(1/n!)

A=\prod_{n=0}^{\infty} \sqrt[n!]{1+\frac{1}{e^{n}}}

\log A=\sum_{n=0}^{\infty} \frac{1}{n!} \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m}\left(\frac{1}{e^{n}}\right)^{m}

\log A=\sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} \sum_{n=0}^{\infty} \frac{(e^{-m})^{n}}{n!}

\exp\left[\sum_{m=1}^{\infty} \frac{(-1)^{m+1}e^{e^{-m}}}{m}\right]=\prod_{n=0}^{\infty} \sqrt[n!]{1+\frac{1}{e^{n}}}

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