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?
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Take $a_n = \frac{1}{n}$ for $n=1,\dots,N$ and $a_n = 0$ for $n > N$. Then both sides of the inequality are $\sim e \log(N)$, hence the sharpness of the constant $e$.
