# Proving the inequality log(1)1!+log2(2)2!+log3(3)3!+⋯>π4\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}

How to prove this inequality?

The left side looks vaguely like the series for $\exp(x)$: the terms starting from $n$th contribute at least as much as the corresponding terms of the series for $\exp(\log n)$. But for the preceding terms, the inequality goes in the opposite direction.