Deduce Carleman's inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } a _ { n }$$
for any sequence $\left( a _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ of strictly positive real numbers such that $\sum a _ { n }$ converges.