Mathematics

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


ANSWER

$$\displaystyle \pi /2$$


SOLUTION
$$\displaystyle \lim _{ n\rightarrow \infty  } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ \sqrt { n^{ 2 }-r^{ 2 } }  } =\lim _{ n\rightarrow \infty  } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ n } \dfrac { 1 }{ \sqrt { 1-\dfrac { { r }^{ 2 } }{ { n }^{ 2 } }  }  } $$

$$\displaystyle =\lim _{ n\rightarrow \infty  } \dfrac { 1 }{ n } \sum _{ r=1 }^{ n-1 } \dfrac { 1 }{ \sqrt { 1-\dfrac { { r }^{ 2 } }{ { n }^{ 2 } }  }  } =\int _{ 0 }^{ 1 }{ \dfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } }  }  } dx$$

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