TITLE: Proving the inequality $\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$
QUESTION [33 upvotes]: How to prove  this inequality?  $$\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$$

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.

REPLY [11 votes]: As @user2345215 and @Mr.G said in the comments, to answer the question as posed is quite simple.
$$
\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots
> \sum_{n=2}^{11} \frac{\log^n(n)}{n!}>0.7855>
 \frac{\pi }{4}
$$