Oh Mr. Fantastic I should've asked you this earlier...I was just soo excited someone actually replied to my post... you see i haven't had much help for a while on mathhelp...but any who. how did you know to compare that sum with $\displaystyle \frac{2^n}{4^n}$?

I believe you've used the comparison test, again, after finding out $\displaystyle \frac{n^5}{4^n}$ converges by the ratio test? Similarly, how did you know to use $\displaystyle \frac{n^5}{4^n}$

Mr F says: Yes, I did.
Thank you Mr. Fantastic