The value of \lim_{n\to\infty}(1/\sqrt{4n^2-1}+1/\sqrt{4n^2-4}+...+1/\sqrt{4n^2-n^2}) is

KVPY SB SX Exam 2011 Question Paper

Question : 16 of 120

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The value of

\lim↙{n→∞}(1/√{4n^2-1}+1/√{4n^2-4}+...+1/√{4n^2-n^2})

Solution:

\lim↙{n→∞}∑↖{n}↙{r=1}1/√{4n^2-r^2}

=\lim↙{n→∞}1/n∑↖{n}↙{r=1}1/√{4 -(r\/n)^2}

=∫↖{1}↙{0}{dx}/√{4-x^2}

=(\sin^{-1}(x/2))^1_0=π/6

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