Does the series converge or diverge the sum as n goes to infinity
n^10/〖10〗^n .
Thank you!
We can try the ratio test.
$\displaystyle \sum_{n=1}^{\infty}\frac{n^{10}}{10^{n}}$
$\displaystyle \frac{u_{k+1}}{u_{k}}=\frac{(n+1)^{10}}{10^{n+1}}\ cdot\frac{10^{n}}{n^{10}}=\frac{(n+1)^{10}}{10n^{1 0}}$
$\displaystyle {\rho}=\lim_{n\rightarrow{\infty}}\frac{(n+1)^{10} }{10n^{10}}=\frac{1}{10}$
$\displaystyle {\rho}<1$
What dies that tell you?.
This is an interesting series. Try to find what it converges to.