$\lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right.$ $+\frac{n}{(n+3) \sqrt{3(2 n+3)}}+\ldots n$ terms $]=\int_0^1 f(x) d x$ then $f(x)=$

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Aicharya · Accepted Answer

Here, $\begin{aligned} & =\lim _{n ightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{n^2}{(n+k) \sqrt{k(2 n+k)}} \ & =\lim _{n ightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{n^2}{(n+k) \sqrt{k(2 n+k)}} 	imes \frac{n^2}{n^2} \ & =\lim _{n ightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{1}{\left(\frac{n+k}{n}ight) \sqrt{\frac{k}{n}\left(\frac{2 n+k}{n}ight)}}\end{aligned}$ $\begin{aligned} & =\lim _{n ightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{1}{\left(1+\frac{k}{n}ight) \sqrt{\frac{k}{n}\left(2+\frac{k}{n}ight)}} \ & =\int_0^1 \frac{1}{(1+x) \sqrt{x(2-x)}} d x=\int_0^1 f(x) d x\end{aligned}$ $f(x)=\frac{1}{(1+x) \sqrt{2 x+x^2}}$

$\lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right.$…

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