$\displaystyle \lim _ {n \rightarrow \infty} {\sum _ {k=1} ^ {n} \frac{2k} {n ^ {2}} \sin ^ {2} \left (\frac{k \pi} {2n} \right)}$ 의 값은?

• ①$\displaystyle \frac{1} {2} - \frac{1} {\pi}$
• ②$\displaystyle \frac{1} {2} - \frac{2} {\pi}$
• ③$\displaystyle \frac{1} {2} + \frac{2} {\pi}$
• ④$\displaystyle \frac{1} {2} - \frac{2} {\pi ^ {2}}$
• ⑤$\displaystyle \frac{1} {2} + \frac{2} {\pi ^ {2}}$