Mathematics

$$\displaystyle \lim_{n\rightarrow \infty }\sum_{r=1}^{n}\frac{1}{n}\sin \frac{r\pi }{2n}$$ is


ANSWER

$$\displaystyle 2/\pi $$


SOLUTION
$$\displaystyle \lim _{ n\rightarrow \infty  } \sum _{ r=1 }^{ n } \frac { 1 }{ n } \sin  \frac { r\pi  }{ 2n } $$

$$\displaystyle=\lim _{ n\rightarrow \infty  } \frac { 1 }{ n } \sum _{ r=1 }^{ n } \sin  \frac { r\pi  }{ 2n } $$

$$\displaystyle =\int _{ 0 }^{ 1 }{ \sin { \frac { \pi x }{ 2 }  } dx } =-{ \left[ \frac { 2 }{ \pi  } \cos { \frac { \pi x }{ 2 }  }  \right]  }_{ 0 }^{ 1 }=\frac { 2 }{ \pi  } $$
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Single Correct Medium Published on 17th 09, 2020
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