I need help understanding an informal principle when doing these equations.

In my book it says that you can discard all but the leading terms. When can you do this?

For example, one problem has sigma from 1 to infinity of (3k^3-2k^2+4)/(k^7-k^3+2)

which can be compared to 3k^3/k^7 or 3/k^4 to check for covergence or divergence

Why can't you get rid of the 3 in the numerator? Basically I don't understand how to use this rule. Can someone explain it in as simple terms as possible?