Answer
First five terms: $0, 2, \frac{3\sqrt{3}}{2}, 2\sqrt{2}, 5\sin(\pi/5)$. The sequence converges to $\pi$.
Work Step by Step
Putting $n = {1,2,3,4,5}$ in ${n \sin \frac{\pi}{n}}$,
$\implies n=1$: $\sin(\pi) =0$
$\implies n=2$: $2\sin(\frac{\pi}{2}) =2$
$\implies n=3$: $3\sin(\frac{\pi}{3}) =3\times\frac{\sqrt3}{2}$
$\implies n=4$: $4\sin(\frac{\pi}{4}) =4\times\frac{1}{\sqrt2}=2\sqrt2$
$\implies n=5$: $5\sin(\frac{\pi}{5}) \approx2.9389$
The limit $\lim_{n\to\infty} {n \sin \frac{\pi}{n}} = \lim_{n\to\infty} {\frac{\sin \frac{\pi}{n}}{1/n}}$, which is of $\frac{\infty}{\infty}$ form. Thus, applying L’Hôpital’s rule,
$$\lim_{n\to\infty} {\frac{\sin \frac{\pi}{n}}{1/n}} = \lim_{n\to\infty} {\frac{(\cos \frac{\pi}{n})(-\frac{\pi}{n^2})}{-\frac{1}{n^2}}}=\pi\lim_{n\to\infty} {\cos0}=\pi.$$
Thus, the sequence converges to $\pi$.
