 So we'll begin our other convergence test with a special test that applies for alternating series, known as the wait for it alternating series test. If we have an alternating series, we can apply the alternating series test. Suppose I have a series that is eventually alternating. If the absolute value of An is eventually decreasing and the limit as n goes to infinity of these absolute values is zero, then our series converges. Otherwise the test is inconclusive. And an important thing to remember is that the alternating series test only applies to alternating series. So if possible, let's see if we can determine whether the series minus 1 to the n over n converges or diverges. The first thing we notice is that the minus 1 to power n will make odd numbered series terms negative and even numbered series terms positive. So this is an eventually alternating series where the eventually is right now. So we can apply the alternating series test. So first of all, we need to determine whether or not the terms of our series are eventually decreasing. Now the minus 1 to power n may make things a little bit more difficult. Fortunately, we only have to worry about whether the absolute value of the terms is eventually decreasing and so that minus 1 to power n can be ignored and so we need to determine if 1 over n is eventually decreasing. So the thing to remember is that if our derivative is negative, then our function is decreasing. So we'll find the derivative and we see that this derivative is eventually actually always negative. So 1 over n is eventually actually always decreasing. Now we'll check out our second requirement. We'll find the limit as n goes to infinity of the absolute value of our series terms. Again, since we're taking the absolute value, that minus 1 to power n just changes the sign so we can drop it out and then find the limit and since our limit is zero, we pass both requirements of the alternating series test and so we know that this series converges. Let's take a look at a different series. This one is minus 1 to power n times a n, where a n will have different expressions depending on whether n is even or odd. However, we might note that a n will always be positive. So the sign of the terms is going to be determined by the minus 1 to power n, which will alternate between positive and negative. So we have an alternating series and we can apply the alternating series test. So we need to compare the absolute values of the sequence terms. Well, since our sequence terms are defined depending on whether n is even or odd, let's consider those two cases. Suppose n is even, then the actual term will be 2 over n. The next term will be for n plus 1. But since that's an odd number, the actual term is going to be 1 over n plus 1 squared. And now we want to compare 1 term to over n to the next term, 1 over n plus 1 squared. Now, we don't know whether that's smaller or larger, so let's leave the inequality unstated and follow the algebra, putting the inequality in at the end. So we can cross multiply and since both n and n plus 1 squared are positive numbers, that won't change our inequality. We'll expand and do a little algebra. And now we have to ask ourselves, self, what is the relationship between 2n squared plus 4n plus 2 and n? Now, this could be larger or smaller depending on the value of n, but remember we're talking about a series. So what we care about is what happens eventually for n sufficiently large. And if n is a sufficiently large number, then 2n squared plus 4n plus 2 is larger than n and that inequality cascades backwards to the beginning. And so if n is even, the next term is smaller. But if n is odd, the corresponding term is going to be 1 over n squared and the next term, where n plus 1 is going to be 2 over n plus 1. And again, we don't know what the relationship is between the two, but we can leave the inequality unstated and do a little bit of algebra. We'll cross multiply and we'll ask ourselves, self, what is the relationship between n plus 1 and 2n squared for n sufficiently large? And we see that eventually 2n squared will be greater than n plus 1 and so 1 over n squared will be less than 2 over n plus 1. So if n is odd, the next term is larger. But this is a problem. The alternating series task requires that our series be eventually decreasing. Since the terms of this series are not eventually decreasing, the alternating series test is inconclusive. And since our series is not eventually decreasing, this is a harder problem and we have to do something much more difficult. And for that, we have to learn a little bit more and pick up a few more convergence or divergence tests.