Answer
\[\frac{1}{10}\]
Work Step by Step
We have to find :-
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^9+\left(\frac{2}{n}\right)^9+...+\left(\frac{n}{n}\right)^9\right]\]
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^9+\left(\frac{2}{n}\right)^9+...+\left(\frac{n}{n}\right)^9\right]=\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=1}^{n}\left(\frac{k}{n}\right)^9\right]\;\;\;...(1)\]
We will use the formula \[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=f(n)}^{g(n)}F\left(\frac{k}{n}\right)\right]=\int_{\lim_{n\rightarrow \infty}\frac{f(n)}{n}}^{\lim_{n\rightarrow \infty}\frac{g(n)}{n}}F(x)\;dx\;\;\;...(2)\]
Using (2) in (1)
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=1}^{n}\left(\frac{k}{n}\right)^9\right]=\int_{\lim_{n\rightarrow \infty}\frac{1}{n}}^{\lim_{n\rightarrow \infty}\frac{n}{n}}x^9\;dx\]
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=1}^{n}\left(\frac{k}{n}\right)^9\right]=\int_{0}^{1}x^9\;dx\]
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=1}^{n}\left(\frac{k}{n}\right)^9\right]=\left[\frac{x^{10}}{10}\right]_{0}^{1}\]
\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\sum_{k=1}^{n}\left(\frac{k}{n}\right)^9\right]=\frac{1}{10}\]
Hence ,\[\lim_{n\rightarrow \infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^9+\left(\frac{2}{n}\right)^9+...+\left(\frac{n}{n}\right)^9\right]=\frac{1}{10}\]
