 In this video, we'll work through a problem to demonstrate the application of the limit comparison test. Let's recall the test. If we consider the series from k equals 1 to infinity of a sub k and another from k equals 1 to infinity of b sub k, where each series contains only positive terms, suppose 2 that rho equals the limit of the quotients a sub k over b sub k as k approaches infinity. If rho is finite and positive, then either both series converge or both series diverge. If rho is equal to 0 and the series from k equals 1 to infinity of b sub k converges, then the series from k equals 1 to infinity of a sub k converges as well. And if rho is infinity and the series from k equals 1 to infinity of b sub k diverges, then the series from k equals 1 to infinity of a sub k diverges as well. Consider the infinite series as shown here. We'll use the limit comparison test to determine whether it converges or diverges. As is the case with the comparison test, we need to identify a series we know the behavior of to compare our series to. We note that as k approaches infinity, the behavior of the numerator will be determined by the leading term 3k cubed. Similarly, the behavior of the denominator will be determined by the term k to the seventh. So our series will behave like the series 3k to the third over k to the seventh in the long term. So that's probably a good choice for us to use. Now we know the behavior of this series. It's a constant multiple of a convergent p series since p is 4, which is greater than 1. So now let's find rho. Rho, which was the limit as k approaches infinity of a sub k divided by b sub k. Well, a sub k was our original general term, so I'll write that here. Remember our goal is to identify whether rho is 0, infinity, or finite and positive. And b sub k, we decided to be 3 over k to the fourth. Cleaning this up, we find we have the limit as k approaches infinity of 3k cubed minus 2k squared plus 4 divided by k to the seventh minus k cubed plus 2 times k to the fourth divided by 3. Expanding this we find the quotient to be 3k to the seventh minus 2k to the sixth plus 4k to the fourth divided by 3k to the seventh minus 3k cubed plus 6. And we know that the behavior is really determined ultimately by these leading terms. So we find that the limit here is equal to 1. We could also verify this using L'Hopital's rule and other approaches. But ultimately, we find that rho is finite and positive. So by the limit comparison test, we know that since the series that we compared to, since this series converges and rho is finite and positive, then so does our original series.