MHB Series Convergence: Ratio Test & Lim. n→∞

[Confusedalways]

I'm trying to determine if $$\sum_{n=1}^{\infty}\frac{{n}^{10}}{{2}^{n}}$$ converges or diverges.

I did the ratio test but I'm left with determining $$\lim_{{n}\to{\infty}}\frac{(n+1)^{10}}{2n^{10}} $$

Any suggestions??

 

[MarkFL]

I would write the limit as:

$$L=\frac{1}{2}\lim_{n\to\infty}\frac{(n+1)^{10}}{n^{10}}=\frac{1}{2}\left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^{10}$$

Can you proceed?

 

[Confusedalways]

Yes thanks!