Nested radical inequality

Let n \in \mathbb{N}. Prove that

    \[\sqrt{2\sqrt[3]{3\sqrt[4]{4\cdots \sqrt[n]{n}}}}<2\]

Solution

The LHS is equal to 2^{1/2}3^{1/6} \cdots n^{1/n!} which by AM – GM is less or equal to

    \[\left( \frac{\sum_{k=2}^n (k/k!)}{\sum_{k=2}^n (1/k!)}\right)^{\sum_{k=2}^n (1/k!)} = \left(1 + \frac{1}{a_n} \right)^{a_n}\]

where a_n=\sum \limits_{k=2}^{n} \frac{1}{k!}. Since a_n \nearrow e-2 <1 it follows from Bernoulli inequality that \displaystyle \left(1 + \frac{1}{a_n} \right)^{a_n} <2.

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