Does $\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$ diverge? Does the limit of this summation diverge?
$$\lim\limits_{n \to +∞} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$
Thanks!

The summand is $$\frac{n \ln{k}}{n^2+k^2} \geq \frac{n \ln{k}}{2n^2} = \frac{\ln{k}}{2n}$$
Therefore the sum, if it exists, is greater than or equal to
$$\frac{1}{2n} \sum_{k=1}^n \ln{k} = \frac{1}{2n} \log{n!}$$
This diverges, by L'Hôpital's rule, expressing $n!$ as $\Gamma(n+1)$.
