$$\lim_{n\to\infty}n^2\left(1-\cos\frac{\pi}{n}\right)=\lim_{n\to\infty}n^2\frac{\left(1-\cos\frac{\pi}{n}\right)\left(1+\cos\frac{\pi}{n}\right)}{\left(1+\cos\frac{\pi}{n}\right)}=\lim_{n\to\infty}n^2\frac{\left(\sin\frac{\pi}{n}\right)^2}{\left(1+\cos\frac{\pi}{n}\right)}=\lim_{n\to\infty}n^2\frac{\pi^2}{\left(1+\cos\frac{\pi}{n}\right)}\left(\frac{\sin\frac{\pi}{n}}{\frac{\pi}{n}}\right)^2$$

Now, the second term above will approach $1$ as $n$ approaches infinitely large. Hence, the final result is $\boxed{\frac{\pi^2}{2}}$

This problem can also be solved using Taylor's expansion.