Mathematics

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

$\displaystyle 2/\pi$

##### 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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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