TITLE: Limit of $(\sum_{k=0}^{n}k^4)/n^5$
QUESTION [5 upvotes]: So i was trying to find this limit:
$$\lim_{n\to\infty}\frac{  \sum_{k=0}^{n}k^4}{n^5}$$
which at first made me think it's zero but soon i realized that it's probably not. I tried expanding that but there's  no $n^5$ in the explansion. Eventialy i tried something like
$$ \sum_{k=0}^{n+1}k^4 - \sum_{k=0}^{n}k^4=(x+1)^4= An^4+Bn^3+Cn^2+Dn+E$$
But again this has no $n^5$ involved.. If someone could provide a hint..

REPLY [2 votes]: Well, what you're writing in the second line is basically T(n), the nth term. To try and find out the summation of the series try expanding the nth term. Having done that you'll find that it decomposes into a 4th power and lesser powers. Sum up the LHS and RHS. You'll automatically get what's going on. 
You did want just a hint. Good Luck.