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Evaluate S_{n}=\sum_{r=0}^{n-1}\frac{1}{\sqrt{4n^{2}-r^{2}}}\: \: as\: \: n\rightarrow \infty

  • Option 1)

    \pi/2

  • Option 2)

    \pi/3

  • Option 3)

    \pi/6

  • Option 4)

    \pi/4

 

Answers (1)

best_answer

As we learnt 

 

Piece wise Continnous Function -

 

Break th function into sub intervals and calculate integrals Separately.

- wherein

If f(x) has discontinnities in \left [ a,b \right ] at finite number of points

 

 S_{n}= \sum_{r=0}^{n-1}\frac{1}{\sqrt{4n^{2}-r^{2}}}= \sum_{r=0}^{n-1}\frac{1}{n\sqrt{4-\left ( \frac{r}{n} \right )^{2}}}

\Rightarrow \int_{0}^{1}\frac{dx}{\sqrt{4-x^{2}}}= \left [ \sin^{-1}\frac{x}{2} \right ]^{1}_{0}= \frac{\pi }{6}

 


Option 1)

\pi/2

Option 2)

\pi/3

Option 3)

\pi/6

Option 4)

\pi/4

Posted by

gaurav

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