I'm not really sure when each of these should be done. In fact, I don't really understand the reason that we use the limit comparison test. Σ1/(n^2+1) So here I can simply say that P=2>1, so the original converges. Σ1/N^3+N^2 Here, I would say that P=3>1, implying the original converges. But my solutions tell me that here I should use a limit comparison with 1/N^3, where the limit ---->N is = 1. So why did I do this? What did I prove by finding the limit that I didn't prove by just comparing directly? What is different in this that I had to use a limit comparison instead of direct? Just the added N^2? I don't understand why. At first I thought it had to do with fulfilling the inequality where An<Bn of the two series if it converges, but in this case, the 1/n^3+n^2<1^n3, so i'm not sure. And I also see that when I have some questions that have similar sums to the one directly above me, they just split the sum and evaluate the p series of both sums to find the divergence; Why does this seem to be an alternative? Thank you, guys.