 Welcome back everyone in this final video for lecture 41 I actually want to introduce the root test, which is like the little brother of the ratio test I say it that way because the three conditions we use for the root test are very similar to that we saw the ratio test And in fact the proof of the ratio test The proof of the roof test pretty much always uses the ratio test So basically the ratio test can prove everything that the root test can prove But there are some situations the root test actually makes to lead an easier limit calculation So we do actually want it as an additional test for us So this time as the name is going to suggest the root test We're going to be looking at the roots of things so particular We're going to look at the sequence where you take the nth root of the absolute value of a n the ratio test We took the limit of ratios the root test We take the limit of roots and we look at what does that limit? What is the sequence converged to what's the limit in that situation? Well, if the limit of the root sequence if that's L And that's greater than one then that actually gives us an absolutely convergent series just like the ratio test If the limit of the roots turns out to be greater than one that actually means the series was divergent In fact, we could have used the test of divergence to determine that and if the the if the sequence of roots Turns out to converge towards the number one then in that situation just like the ratio test It's inconclusive so let's jump immediately to an example One example where the root test actually does turn out to be a little bit more effective than the ratio test would be the series We would take n equals one the sum of n equals one to infinity of the Sequence 2n plus 3 over 3n plus 2 all raised to the nth power and so while the root test Why the root test is a little bit nicer because everything is raised to the nth power right here Ratios could work right here But as we have a rational expression raised to an exponent if you took ratios here gonna have fractions inside of fractions It's gonna be quite messy, but the root test is actually super clean in this situation If you take the nth root of this term the absolute value of 2n plus 3 over 3n plus 2 All raised to the nth power right here the root the the nth root is gonna cancel the nth power right here And we end up with the absolute value of 2n plus 3 Over 3n plus 2 and if we take the limit here because this is a balanced rational function We have a 2n on top a 3n on the bottom This is going to converge towards the limit two-thirds taking absolute values there now this limit value L This is the limit of the root sequence. This is less than one and so this tells us that our series is going to fact be Absolutely convergent by the root test and so the root test works out really nicely here Alternatively, if we wanted to use the ratio test we get 2n plus 3. Sorry back up a little bit We're gonna get 2n plus 1 plus 3 over 3 in plus 1 Plus 2 raised to the n plus 1 power and this will sit above 2n plus 3 over 3n plus 2 to the nth power and while you can argue that this limit will converge to something L and that L value be less than 1 you could use the ratio test to determine that this thing is Convergent you could do that But if you compare the two side-by-side you can see that the root test is a much cleaner argument here And so it's nice to have the root test as an alternative to the ratio test Especially you have nice powers that will cancel out with the in route with that that comes with the root test here