The following formulae are used in finding the limits as integration:
If $f$ is continuous on [0,1], then $\lim_{n \to \infty} \frac{1}{n} \Sigma_{k=1}^n f( \frac{k}{n})=\int_0^1f(x)dx.$
If $f$ is continuous on [a,b], then $\lim_{n \to \infty} \frac{b-a}{n} \Sigma_{k=1}^n f( a+\frac{k(b-a)}{n})=\int_a^bf(x)dx.$
Problems:
1. Evaluate $\lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{4 n^{2}-1}}+\frac{1}{\sqrt{4 n^{2}-4}}+\frac{1}{\sqrt{4 n^{2}-9}}+\ldots \cdot \frac{1}{\sqrt{3 n^{2}}}\right]$
Ans: $\left.\sin ^{-1} \frac{x}{2}\right|_{0} ^{1}=\frac{\pi}{6}$.
2. [JEE] Find the limit $\lim _{n \rightarrow \infty} \frac{1^{p}+2^{p}+3^{p}+\ldots+n^{p}}{n^{p+1}}$.
Ans: $\frac{1}{p+1}$
If $f$ is continuous on [0,1], then $\lim_{n \to \infty} \frac{1}{n} \Sigma_{k=1}^n f( \frac{k}{n})=\int_0^1f(x)dx.$
If $f$ is continuous on [a,b], then $\lim_{n \to \infty} \frac{b-a}{n} \Sigma_{k=1}^n f( a+\frac{k(b-a)}{n})=\int_a^bf(x)dx.$
Problems:
1. Evaluate $\lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{4 n^{2}-1}}+\frac{1}{\sqrt{4 n^{2}-4}}+\frac{1}{\sqrt{4 n^{2}-9}}+\ldots \cdot \frac{1}{\sqrt{3 n^{2}}}\right]$
Ans: $\left.\sin ^{-1} \frac{x}{2}\right|_{0} ^{1}=\frac{\pi}{6}$.
2. [JEE] Find the limit $\lim _{n \rightarrow \infty} \frac{1^{p}+2^{p}+3^{p}+\ldots+n^{p}}{n^{p+1}}$.
Ans: $\frac{1}{p+1}$
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