 Let's determine the convergence of the given series. Let's take the sum as n equals one to infinity of the sequence two to the n over n factorial. Now take a moment right now to contemplate, maybe for a minute, which convergence test would you use to determine the convergence or divergence of this series? And maybe work out the calculations that go with those convergence tests. Many of them require an anti-derivative or limit calculation. Pause the video right now and do that. Now, if it was me, when I look at this series, I can't help but see there is a factorial in the denominator. Factorials to me nearly always mean I want to use the ratio test. Now, if you look at the numerator, you have two to the n. Exponentials work great for the ratio test. So the ratio test is how I would want to do it. I don't know an anti-derivative for n factorial, but I don't need anti-derivatives here because I can use the ratio test. So if you look at your ratio of consecutive terms, you have a n plus one over a n, to the absolute value of this, what I'm gonna do is I'm gonna first write the a n term. So that's gonna look like two to the n plus one over n plus one factorial, right? So that's the first thing I wanna do, multiplication there. And so this is the a to the n plus one term. The next thing I wanna do is I wanna write the one over a to the n term. The reason I'm kinda segregating these things is because as they're both fractions, I'm gonna get fractions instead of fractions, I'm avoiding that. So I'm just gonna write the denominator here as just multiplying by the reciprocal. So if I take one over a to the n, I just take it's reciprocal, you can n factorial on top, and then two to the n right here. Now, what often happens with the ratio test is you're gonna see that when you compare terms from the a n plus one versus the one over a to the n, there's gonna be some things that look kinda similar, right? Like there's an exponential here and there's an exponential here. Likewise, there's a factorial here and a factorial here. If we wanna kinda regroup those so that these like functions are together. So we have this two to the n plus one on top and this two to the n on the bottom. Then we're gonna get an n factorial on top and an n plus one factorial on the bottom. And so now we wanna simplify these fractions using their factorizations. Now, two to the n plus one factors as two to the n times two to the first. So two to the n's cancel, we're left with a two, which I'm gonna record that here. Then with the n factorial and the n plus one factorial, factorials have a very nice factorization principle. You're gonna get n plus one times n factorial through which the n factorials cancel and we're left with an n plus one on the denominator. So this is just the sequence a n plus one over a n. I dropped the absolute values because n will be positive here. So everything is positive in question. So now we wanna take the limit here. We're taking the limit as n approaches infinity here. So as n approaches infinity, we'll end up with something like two over infinity which squashes down to zero. Now this is the limit of our ratio sequence. The ratio test tells us that if this limit is less than one, then the original series was in fact convergent. And in fact, we can say that this thing is absolutely convergent, but given that two to the n and n factorial are positive things, absolute convergence doesn't really tell us anything we didn't already know, but this is an absolutely convergent series by the ratio test. And I think that's probably the cleanest, the cleanest one to do here.