L-10

\(10).\;\; \displaystyle S_{n}=\frac{1}{\sqrt{4 n^{2}-1^{2}}}+\frac{1}{\sqrt{4 n^{2}-2^{2}}}+\cdots \frac{1}{\sqrt{4 n^{2}-n^{2}}}\)

SOLUTION

\(\displaystyle S_{n}=\sum_{k=1}^{n} \displaystyle \frac{1}{\sqrt{4 n^{2}-k^{2}}}\)

\(\displaystyle \Rightarrow S_{n}=\sum_{k=1}^{n} \displaystyle \frac{1}{n \sqrt{2^{2}-\displaystyle \frac{k^{2}}{n^{2}}}}\)

Using Riemann sum Relationship to Definite Integral We will get

\(\displaystyle \Rightarrow \quad \int_{0}^{1} \frac{d x}{\sqrt{2^{2}-x^{2}}}\)

\(\displaystyle \Rightarrow\left|\sin ^{-1} \frac{x}{2}\right|_{0}^{1}\)

\(\displaystyle \Rightarrow \sin ^{-1}\left(\frac{1}{2}\right)\)

\(\displaystyle =\frac{\pi}{6}\)

ANSWER : \(\displaystyle =\frac{\pi}{6}\)