 In this video, we provide the solution to question number eight for practice exam number four for math 1220 We're given a sequence a sub n which is given by 2n squared plus n plus 3 all over 4n cubed minus the square root of n plus 1 and so we're interested in what is the convergence of the associated infinite series So we take the sum where n ranges from 1 to infinity of the sequence a and we just talked about And we have to determine whether the sequence the series is convergent or divergent using the limit comparison test and Absolutely, I'd want to use the limit comparison test because this thing is a nasty-looking bugger And so who would we compare it to so there's some sequence bn that we'd compare it to such that the Limit of the ratio a n over b n should equal one and so in order to get exactly one We basically just want to look at the leading terms there So looking at looking at a sub n right here Who are the leading terms on the top the dominant term the fastest growing term is the 2n squared in the denominator? The dominant term is going to be the 4n cubed so b sub n is just going to be the leading terms there 2n squared over 4n cubed we want to simplify that 2 goes into 4 2 times n squared goes into n cubed n times so this simplifies to be 1 over 2n and So that is the sequence we'd be using so bn is equal to 1 over 2n and And that's because of that we get that the series The series of a n will have the same convergence as the series of the bn and now the series bn The series 1 over 2n here. This is divergent By the p-test. I mean, it's just one half times. It's one half times the Harmonic series and so since these two series will have the same convergence This series likewise will be divergent by the limit comparison test