 Let us determine the convergence of the given series, the sum where any ranges from one to infinity of negative one to the n times n cubed over n to the fourth plus one. So you watching the video, I would actually recommend that you take a moment, a minute or two and contemplate what convergence tests what I use to determine the convergence of this series or it's divergent. Pause the video if necessary and try to do that right now. For me, I would, I see this negative one to the n right here and this automatically tells me that, we have an alternating series at our hand here, an alternating series. And so as such, since it's an alternating series, this makes me inclined to use the alternating series test. The alternating series test is fairly simple to check and thus we can find convergence pretty quickly. So if we take the absolute value of the sequence in play, we're gonna take bn to equal n cubed over n to the fourth plus one. Well, that sequence is clearly positive because we're taking the absolute value, we're just ignoring the negative one to the n there. Is it continuous? Yep, it is. Is it decreasing towards zero? That's the real part we need to worry about here. Is this thing decreasing towards zero? Well, going to zero is not so hard to see here because if we look at just the leading terms here, that determines the asymptotic behavior, this thing will be approximately the same thing as n cubed over n to the fourth, which simplifies to be one over n, which this does go down to zero, all right? Notice right here, this sequence, this one over n, it's actually gonna play an important role in just a little bit, but we can see very quickly that the terms in sum are converging towards zero and this does happen in a decreasing manner. If we think of the graph of this thing, if we want to think of some type of continuous extension of this graph, you have the x-axis right here. What our sequence is doing is that it's gonna be decreasing from the positive side towards zero like this. And honestly, if it wiggles, if it's got the wiggles, that's like bananas in the pajamas or something like that, then it doesn't really matter, this function is not actually doing that, but it doesn't really matter. This function as it asymptotically approaches the x-axis, ultimately it's gonna have to reach a point where it is decreasing towards zero. There's no type of oscillation going on here because we have purely just this algebraic rational function. Rational functions have a limited number of oscillations and therefore eventually whatever craziness happens, it will eventually have to decrease towards zero. So we get something like this right here. So that tells us that the alternate series test applies and the alternate series test will tell us that this series is convergent. That's good to know. It's convergent. That makes us happy, it answers the question. But what if we want a little bit more specificity here? What if we don't just wanna know whether it's convergent? Is this thing absolutely convergent or conditionally convergent? Now, because it's an alternate series, right? Not all the terms in this series are positive. And so oftentimes we show that things are convergent and they're automatically absolutely convergent because every term in the sequence is positive. We do this when we use like the integral test or comparison test. The series, all the terms there are positive, so for alternate series, not all the terms are positive so it could be only conditionally convergent. How would we know about that? Well, if we wanna find out more, we have to investigate the absolute series. So if we take the absolute value of these things, sorry, the absolute value of the series here, what happens in that situation? Well, that's the sequence we were considering a moment ago, right? So if we switch to the absolute series, we take the sum where n equals one to infinity, we're gonna get n cubed over into the fourth plus one. In the process of trying to determine that this sequence went to zero, so it's not in violation of the test of divergence, we saw that this thing was comparable to this sequence, dun dun dun, which that means this series will be limit comparable to the series one over n, that is the harmonic series. So this series right here, the harmonic series, the harmonic series is divergent. This is either just because we remember the harmonic series, it shows up a lot. It's divergent by the P test if we need a specific reference there, P equals one. Now the divergence of the harmonic series shows that the absolute series is likewise divergent. And we're in this case using the limit comparison test. We're using the limit comparison test to show that the absolute series is divergent. So this now has shown us that our series is convergent, but it's not, it's not absolutely convergent. And so thus we must resolve this with the final answer that our series is conditionally convergent. And so this is a good example to remember here because as I've taught students in the past, it turns out that they find conditional convergence as one of the hardest ones to show because showing that something's conditionally convergent means you show it's convergent, but not absolutely convergent. And that actually turns into two problems. We have to first show that the series is convergent, which we did using the alternating series test, but then we have to show that it's not absolutely convergent. And we did that using the limit comparison test when we looked at the absolute series right here. And so there's kind of like two series arguments that have to be made in this situation. So I kind of wanna throw this example as an important example of showing that a series could be conditionally convergent. Showing absolute convergence is often easier because many of the convergence tests we know, the ratio test, the root test, the integral test, the comparison test, the limit comparison test, those will show absolute convergence, the geometric series test, P series, those will give us absolute convergence. And so for this lecture series, math 1220 here, if something's conditionally convergent for the most part, it'll probably be an alternating series, an alternating series where the sequence of positive terms is asymptotic to one over n, or it's asymptotic to some P series that's divergent. Again, that's not an absolute, but for this lecture series, those are generally the examples of conditional convergence that we find. And so I wanted to point that out for everyone.