We want to use the interval test to determine whether this Siri's converters or diverges. So they tell us to check the conditions to ensure that internal testes satisfied. So let's first go ahead and check to see if our function is continuous. Our that's positive is the first thing we should actually check. So let's go ahead and call. No, our sequence there. So natural log of in squared over in. Well, first, we could rewrite this us two times the natural log of in over in. And I'm just gonna want to rewrite it like this for when we check to see if it's decreasing car, not some when we take the derivative of it. Well, if we're plugging in values larger than two natural log into is positive in is also going to be positive. Divide to positive numbers. So this is going to be strictly greater than zero. Now we need to check to see if the function is continuous and the only issues will have is when we divide by zero. So since and his large and the two no issues there and for negative values, because remember, the natural law could only plug in numbers larger than zero and since and is larger than zero, it's also good. So to continues, functions divided by each other at least on the interval, larger than two works out. And now we need to check to see if this is decreasing. So we will go ahead and find the derivative of this so f prime. So I'm gonna take the derivative of this one over here so we can use quotient rule. To do that, there's gonna be two times so low d high. So one over, n minus high de lo all over the square. What's below? And now, if we were to go ahead and simplify this down would have two times one meister natural log of and all over and squared. And remember, this is for in greater than or equal toe to so we can make sure that this will be negative if our numerator it's negative. So this is lessons because in squares, always positive. So we would add natural log cabin over. So we have one greater than natural log of in, and then we exponentially it inside. So we want to make sure that in is larger than E Qari and so he is about 2.71 So that means if we take in to be three, then this will work so we can go ahead and just let nd three. And in this case, well, we can come over here and rewrite our Siri's to be. So we first plug in to so I'd be natural log of or over two and then some from and is equal to three to infinity of natural log of in I'll just due to natural event over in. And now this is still satisfying it being positive and continuous. Because on this interval, here is where it's positive and as long as we have in three here, then we have that it's decreasing on this whole interval. So this here will check out. So now we can go ahead and apply the integral test toe. This Siri's here and so is going to be three to infinity of. I'm gonna factor that two out so two times the natural law given over and now we can integrate this using a u sub and it looks like the substitution we're gonna want to make is gonna be you. Is he going to the natural log of it. And so then D you is going to be won over end Deion. So then that means we can go ahead. Every right are integral here as two times the integral oh so natural lago three and then the limit, as in a purchase. Affinity. Ah, natural log of in that should go to infinity still So our upper bound is still gonna be infinity and then we just have you, d you know, interpreting that we need to power rule So we'd have you squared divided by two. So that two out there who can't sell so we'd have you squared evaluated from the natural log of three Have you squared, Evaluate from natural of the three to infinity. And so if we were to go ahead and write This is the limit though, So the limit as be approaches infinity. So remember replacing infinity here with B So the B B squared minus the natural log of three squared Well, b squared diverges to infinity Natural log of two squared Our national August 3 squared is just a constant so that is a matter So this diverges and so that implies that are Siri's and is he going to to infinity of the natural log of in squared over in diverges as well?