Let $(a_j)_{j\ge 1}$ be a sequence of positive real numbers. Carleman's inequality 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?

Let $(a_j)_{j\ge 1}$ be a sequence of positive real numbers. Carleman's inequality 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?

Let $(a_j)_{j\ge 1}$ be a sequence of positive real numbers. Carleman's inequality 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?

Let $(a_j)_{j\ge 1}$ be a sequence of positive real numbers. Carleman's inequality 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?