I came across the following series on a test today: $$\sum_{n=1}^{\infty}\frac {(n^3-n^2+3)(n^{5/3})} {n^5+10^{10}n^4-1}$$ The question was to figure out whether this converged absolutely, conditionally, or diverged.
Well, to start off I tried distributing the $(n^{5/3})$ and then rewriting the series with the exponents in fractional form. So it was $$\sum_{n=1}^{\infty}\frac {(n^{14/3}-n^{11/3}+3n^{5/3})} {n^{15/3}+10^{10}n^{12/3}-1}$$
I figured that it probably diverges because it "looks like" the p-series $\sum_{n=1}^{\infty}\frac 1 {n^{1/3}}$ but I didn't know how to test it further and actually show that it diverges. So after that I was stuck!
Can someone please help me understand how to do this problem?