Here's the problem:

$\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^n4^nx^n}{\sqrt{n}+10}$

so I use ratio test (obviously):

$\displaystyle \lim_{n\rightarrow \infty}\left | \frac{(-1)^{n+1}4^{n+1}x^{n+1}}{\sqrt{n+1}+10}\times \frac{\sqrt{n}+10}{(-1)^n4^nx^n}\right |$

after some cancellations I have:

$\displaystyle \lim_{n\rightarrow \infty}\left | \frac{-4x(\sqrt{n}+10)}{\sqrt{n+1}+10}\right |$

It seems that I could say that $\displaystyle \sqrt{n}$ cancels, but then again, n + 1 will increase faster than n.

I can't figure out where to go from here though. Help? Thanks!