This is incorrect. Dividing both numerator and denominator of $\displaystyle \frac{n^2}{n^2+ k^2}$ by $\displaystyle n^2$ gives $\displaystyle \frac{1}{1+ \left(\frac{k}{n}\right)^2}$.

Oops! I totally missed that typo...

The problem is that k is the one going to infinity...I did a mess with the indexes.

Oh, well...

Tonio
And, now, unfortunately, that is NOT the same as $\displaystyle \frac{1}{1+ x^2}$.

Perhaps dividing both numerator and denominator by $\displaystyle k^2$ giving $\displaystyle \frac{\left(\frac{n}{k}\right)^2}{1+ \left(\frac{n}{k}\right)^2}$ so we can think of it as a Riemann sum for $\displaystyle \int \frac{x^2}{1+ x^2} dx= \int x-\frac{x}{x^2+ 1}$ will work.