Alternating series test

- [Voiceover] Let's now expose ourselves to another test of conversions, and that's the alternating series test. And I'll explain the alternating series test, and I'll apply it to an actual series while I do it to make the explanation of the alternating series test a little bit more concrete. So let's say that I have some series, some infinite series. Let's say it goes from n equals k to infinity of a sub n. Let's say I can write it as, or I can rewrite a sub n. So a sub n is equal to negative one to the n, times b sub n, or a sub n is equal to negative one to the n plus one, times b sub n, where b sub n is greater than or equal to zero for all the n's we care about. So for all of these integer n's greater than or equal to k. So if all of these things are true, and we know two more things. And we know number one, the limit as n approaches infinity of b sub n is equal to zero. And number two, b sub n is a decreasing sequence. Then that lets us know that the original infinite series is going to converge. So this might seem a little bit abstract right now. Let's use this with an actual series to make it a little bit more concrete. So let's say that I had the series from n equals one to infinity of negative one to the n over n. And we can write it out just to make this series a little bit more concrete. When n is equal to one, this is going to be negative one to the one power. Actually, let's just make this a little bit, let's make this a little bit more interesting. Let's make this negative one to the n plus one. So when n is equal to one, this is going to be negative one squared over one, which is going to be one. And then when n is two, it's going to be negative one to the third power, which is going to be negative one half. So it's minus one half, plus one third, minus one fourth, plus, minus, and it just keeps going on and on forever. Now, can we rewrite this a sub n like this? Well sure, the negative one to the n plus one is actually explicitly called out. We can rewrite our a sub n, so let me do that. So a sub n, which is equal to negative one, to the n plus one over n, this is clearly the same thing as negative one to the n plus one times one over n. Which we can then say, this thing right over here could be our b sub n. So this right over here is our b sub n. And we can verify that our b sub n is going to be greater than or equal to zero for all the n's we care about. So our b sub n is equal to one over n. Now clearly this is going to be greater than or equal to zero for any positive n. Now what's the limit? What's the limit of b sub n as n approaches infinity? The limit of one over n as n approaches infinity is going to be equal to zero. So we satisfy the first constraint. And then, this is clearly a decreasing sequence. As n increases, the denominators are going to increase, and with a larger denominator you're going to have a lower value. So, we can also say one over n is a decreasing sequence for the n's that we care about. So this is satisfied as well. So based on that, this thing right over here is always greater than or equal to zero. The limit as one over n or as our b sub n, as n approaches infinity is going to be zero, it's a decreasing sequence. Therefore we can say that our original series actually converges. So n equals one to infinity of negative one to the n plus one over n. And that's kind of interesting, because we've already seen that if all of these were positive, if all of these terms were positive, we'd just have the harmonic series. And that one didn't converge, but this one did. Putting these negatives here do the trick. And actually, we can prove this one over here converges using other techniques. And maybe if we have time... Actually, in particular the limit comparison test, I'll just throw that out there in case you are curious. So this is a pretty powerful tool. It looks a little bit like that divergence test, but remember the divergence test is only useful if you want to show something diverges. If the limit of your terms do not approach zero, then you say okay, that thing is going to diverge. This thing is useful because you can actually prove convergence. Now once again, if something does not pass the alternating series test, that does not necessarily mean that it diverges, it just means that you couldn't use the alternating series test to prove that it converges.