1\lim_{n\to\infty}\frac{(n+1)^{9}+(n+2)^{9}+\cdots+(n+n)^{9}}{1^{9}+2^{9}+\cdots+n^{9}} =2^{k}-1 \\ \\ \lim_{n\to\infty}\frac{\frac{(n+1)^{9}+(n+2)^{9}+\cdots+(n+n)^{9}}{n^{10}}}{\frac{1^{9}+2^{9}+\cdots+n^{9}}{n^{10}}} \\ \lim_{n\to\infty}\frac{\sum_{r=1}^{\infty}{\frac{1}{n}\left( 1+\frac{r}{n}\right)}^{9}}{\sum_{r=1}^{\infty}{\frac{1}{n}\left( \frac{r}{n}\right)}^{9}}\\ \frac{\int_{0}^{1}{\left(1+x \right)^{9}\mathrm{dx}}}{\int_{0}^{1}{\left(x \right)^{9}\mathrm{dx}}} \\ (2)^{10}-1
thank u che for this ;)
