TITLE: inequality proof
QUESTION [8 upvotes]: how could we show that the following holds for $k \ge 2$, $k \in \mathbb{N}$:
$$ 1 < \left( k - \frac{1}{2}\right) \log{\left(\frac{k}{k-1}\right)} $$
or equivalently, how can we show that
$$ e < \left( \frac{k}{k-1} \right)^{k-1/2} $$
Thanks!

REPLY [4 votes]: An article on this problem appeared in the American Mathematical Monthly 2 years ago. It contains 3 proofs. The reference is:

Amer. Math. Monthly 117 (2010) 273–277. doi:10.4169/000298910X480126

You can view the article here.