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!

$$\sum_{k\leq n}\frac{n\log\left(k\right)}{n^{2}+k^{2}}\geq\frac{1}{2n}\sum_{k\leq n}\log\left(k\right)
 $$ and by partial summation we can see that $$\sum_{k\leq n}\log\left(k\right)=n\log\left(n\right)-\int_{1}^{n}\frac{\left\lfloor t\right\rfloor }{t}dt\geq n\log\left(n\right)-n+1
 $$ where $\left\lfloor t\right\rfloor $ is the floor function, so the sequence diverges.
