Let $(a_j)_{j\ge 1}$ be a sequence of positive real numbers. [Carleman's inequality][1] says that
$$
\sum_{n\ge 1}\left(\prod_{1\le j\le n} a_j\right)^{1/n}< e\sum_{n\ge 1} a_n.
$$
The constant $e$ is optimal. What is the simplest (or a simple) choice of $a_j$ to check that optimality?

[1]: https://en.wikipedia.org/wiki/Carleman%27s_inequality