 Let us determine the convergence of the series where we add together from n equals one to infinity sine of n over factorial. Hmm, this one's kind of a tricky one and I want you to, I want you watching this at home to kind of pause the video for a moment and actually think about what would I do to determine the convergence of this one? And I want you to be disciplined that I don't want you to unpause this video and finish watching until after you have a good idea on what to do. Hey, I told you not to unpause the video until you figured this one out on your own here. At least you gave it a good, honest try here. So you pause that video and do it for real. All right, so assuming that we all took a good stab at this one, we might have, some of us might have been successful, some of us might have got stuck because like I said, this one is a tricky one, right? So what's going on here? This, really the obstruction is coming from the sine of n on the top because unlike many series, this is not a positive series, right? Sine of n, right, this is sometimes positive but it's also sometimes negative. Potentially for different choices of sine, this thing will be jumping between numbers, we'll be choosing real numbers between negative one and one. But I'll never, so it's not a positive sequence. It's not a positive sequence. And so that's an important thing to make mention here because the fact that it's not positive means that we can't use the integral test, even if we could find the anti-derivative sine of n, even if you could find the anti-derivative sine of x over x factorial, whatever x factorial even means, but the integral test does require that this thing become eventually a positive function. It also has to be eventually continuous and eventually decreasing, but sine of n will never eventually be positive because it will be equally likely negative as it is positive. So the integral test doesn't apply here. Neither does the comparison test. The comparison test, and I'm including the limit comparison test here as well. The limit comparison test only works when we compare a positive sequence to a positive sequence. And then when there's an appropriate comparison there, we can then say something about the convergence of the series. So the comparison test and the integral test does not apply here. Now oftentimes we could use the alternating series test but this series is not an alternating series. It's not alternating series, right? Which that tells us that the alternating series test doesn't apply here as well. Now an alternating series does one of two things. It either goes positive, negative, positive, negative, positive, negative, positive, negative, or maybe it starts negative, positive, negative, positive, negative, positive. You have this back and forth every other time. That's not what this series will do. This series will do something like positive, positive, negative, positive, positive, positive, negative, negative, positive, negative, positive, or something like that, right? It's not an absolute every other time. So a lot of the convergence tests that we know about don't apply. It's clearly not a P test because of the sign of N on the top. It's not a geometric series so that doesn't apply. So we're actually showing a lot of options that we can't do. We cannot use these things, all right? So again, this is getting super depressing here. What is left? What could we still use? Well, we potentially have the ratio test, don't we? The ratio test, it doesn't have to be a positive or alternating series. So potentially have the ratio test. But if we were to use the ratio test, what are we gonna have? We're gonna be looking at the sequence A N plus one over A N. And so that would look something like the following. Sign of N plus one over sign of N times N factorial over N plus one factorial. Now the factorials aren't a problem for the ratio test. This would just simplify to be one over N plus one. But this creature right here, sign of N plus one over sign of N. If we wanted to determine whether that thing was, what's the limit there, right? What's the limit as N goes to infinity of sign of N plus one over sign of N. The problem is sign of infinity is not defined. There's no limit there because of the oscillating nature of the function. So one has some concerns about what's gonna happen here, right? What is gonna happen to this thing? Now, this limit calculation is not gonna be super helpful for us if we try to resolve it, right? Now again, we actually have to calculate in with the idea there's an N plus one on the bottom. That does actually give me a little bit of hope that this thing right here, right? Sign of N plus one over N plus one. Maybe that'll give us a home up, oh wait, that's if N goes to zero, right? That equals one. If N goes to infinity, feels like something's going on here. The issue is one could make the argument that since sign is bounded above, it's always between negative one and one like we saw before. The denominator is gonna get bigger and bigger and bigger in terms of absolute value, although there's some oscillation going on here. This thing, we could try to make an argument that this thing is gonna go off towards zero maybe, right? One could do that, but that's gonna be kind of a difficult limit to do for a calculus student, possible, but we could perhaps make that work. We could also try the root test. So I guess what I'm saying is the ratio test might have some promise, but it's gonna be a difficult limit. One could try the root test instead, but you're gonna see that in my opinion, the root test might actually be somewhat worse in this situation, right? Because we have to take the nth root of the absolute value of sign N, N factorial. That in my opinion looks like it's gonna be even worse. How do you deal with the nth root of sign? So no, the ratio test, if you're gonna go this direction, the ratio test is something you could do. But there's another, I think, much easier thing to show here that some people often forget about, okay? So one thing to remember is that if a series is absolutely convergent, then it's convergent, right? Absolute convergence, this implies convergence. And why this is worth considering is the following idea. If we take the absolute series, so we take the sum where n equals one to infinity, we're gonna get the absolute value of sign of N over N factorial. Now N factorial is always positive. So in terms of the absolute value, that's only the absolute value of sign, right? And so when we take this absolute value sequence, this absolute sequence, one thing I wanna mention now is that this is now a positive sequence. So some of the tools that we had to throw out earlier actually applicable to the absolute series. For example, the integral test applies now. I don't wanna do the integral test because again, this would be a horrible anti-derivative. But maybe a comparison test would actually work, right? That's kind of important to know. The absolute value of sign of N, this will be bounded below by zero, since sign, of course, since we're taking average value as positive now, but it's also bounded above by one. And so we could then compare this to the series one over N factorial where N equals one to infinity, which this is a series that we have looked at before. This is one that you're gonna wanna know. I mean, this series right here, well, you could use the ratio test very simply to do this series is what I guess I'm trying to say here. For this one, your limit here, you're looking at this N plus one, I guess it's actually, the limit you're looking at here would be N factorial over N plus one factorial, which that simplifies to be one over N plus one, which that converges towards zero here, which is less than one. So the ratio test would apply convergence here. Ratio test implies convergence here. And then the comparison test implies convergence here. And since this is the absolute series, the convergence of the absolute series implies the convergence of the original series. And so this shows that the original series is convergent. And if I had a quote, a test here, we'll call this by the test, the test of absolute convergence because absolute convergence is an easier thing. And one reason we like absolute convergence is when something is positive, the series is entirely positive, we have more options to the test, it's convergence or divergence than if something is sometimes positive, sometimes negative. So there is an advantage by looking at the absolute series in this case. And that's actually the approach I would recommend. You could have started off with the ratio test without using the absolute series. And then actually give us the absolute convergence of the series, which is great. That is good to know here. So kind of compare these different techniques here. Again, for your typical student absolute convergence is the direction to go here. I think that'll be much more helpful. But when you're approached with a series and no one's telling you which convergence test to use, remember, this is a street fight. We can use whatever tools, whatever weapons we have available for the fight. And so the only rule is that if you can do it, then you can do it. And so try to think through these problems, try to try different approaches. Some will be successful, some won't. You could try to analyze it like we did in this video. Why can't I use this tool? Why could I use this one? And as you try to argue with yourself with these, you'll really find a deeper understanding of the series and why they converge and why they sometimes diverge.