 Welcome back to our lecture series, Math 1220, Calculus II for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Mistledine. This is our first video for lecture 41, for which we're going to talk about the ratio and the root test, the ratio test being the foremost important thing in this lecture 41. These are going to be our final convergence test that we're going to top out for series. In order to preference the ratio test, we're going to talk about this idea called absolute convergence. Suppose that we have a series, you take the sum of the sequence a sub n, we say that this series is absolutely convergent. If it's absolute sequence, I should say if the absolute series is convergent. We had actually talked about the absolute sequence earlier when we talked about the alternate series test. The absolute sequence is you take your sequence and you take absolute values of everything. The absolute series would then take the sum of the absolute sequence. We say that a series is absolutely convergent if its absolute series is convergent. Hence why we call it absolutely convergent. Now, let's look at some examples of such a phenomenon. Consider the series where we take from n equals 1 to infinity, the sum of negative 1 to the n minus 1 over n squared. Notice that this is an alternating p series. We start off with 1, then the second term's negative, then the third term's positive, and the fourth term's negative, and you'll alternate back and forth, back and forth, back and forth. This is an alternating series. If we take absolute values of these terms, like we did here, well, then that negative 1 is just going to poof. It just disappears, and we're left with the sequence 1 over n squared. If you add those terms together, the sum n equals 1 to infinity of the sequence 1 over n squared, then you'll get 1 plus 1 fourth plus 1 ninth plus 1 sixteenth, et cetera, et cetera, et cetera. This will then be a p series, a p series where p equals two, which is thus convergent by the p test. So this shows us that the absolute series is convergent, which implies that the original series is absolutely convergent. One thing I should mention that if you have a positive series, that is every term in the series is positive, convergence and absolute convergence will be the exact same thing, right? You potentially could get a series because it has some negative terms. Well, it's absolute series with something different, and therefore the convergence might be, you know, a different question, Han, right? Take for example, the alternating harmonic series, right? We've seen this one before when we talked about the alternating series test. This is an alternating series. It's alternating harmonic, so it goes 1 minus 1 1⁄ plus 1 3rd minus 1 4th, et cetera, et cetera. If we take the absolute series right here, then we're gonna get the harmonic series, which we know is divergent. So because this series, if you look at its absolute series, its absolute series is divergent. So this series is not, it's not absolutely convergent. But on the other hand, we did see that the alternating harmonic series by the alternating series test is convergent. So we see that in this series, it's convergent but not absolutely convergent. So the two things are measuring slightly different things. And so to describe this, if a series is convergent, it's convergent but it's not absolutely convergent, then we say the series is conditionally convergent. It's sort of a play on words right there because absolute convergence doesn't actually mean like, it's guaranteed, brother. It just means that the series of absolute, the series of absolute values is convergent well. The opposite of being absolute is conditional. So a series is conditionally convergent if it's convergent but not absolutely.