 So let's say if we can answer this question of whether or not an infinite series converges or diverges. Now let's talk a little bit about the anatomy of a series. We can split any series into a head and a tail. What that means is we can split off the beginning portion of the series, that sum from n equals 0 up to any value k, plus the rest of the series, where we can make this split at any k. Now, since this head consists of a finite number of terms, it follows that for any series, the head will always be finite. So the convergence or divergence of a series depends on the convergence or divergence of the tail. Because of this, we can, for purposes of convergence testing, ignore where the series starts. One other idea that's useful is that it's easiest to deal with something that is always but very few things are always, but there's good news. We might be able to consider things that are eventually. And this is important when dealing with infinite series, because we don't have to worry about what happens at the start at the head of the series. We only have to worry about what happens in the tail of the series. We only have to worry about what happens eventually. For example, suppose I want to show that the terms of this series are eventually decreasing. So how can we do that? Remember to determine whether a function is increasing or decreasing, look at the sign of its derivative. So our series has terms that are produced by the function n to the power 100 over 1.01 to the power n. So let's differentiate this. And the sign of the derivative is going to depend on the sign of this fraction. So we know that a fraction will be negative if numerator and denominator have opposite signs. But the denominator 101 to the power 2n is a power, so it will always be positive. So if we want this to be a decreasing function, we'll need the derivative to be negative. And so the question is, when will the numerator be negative? So we'll set up our inequality and we'll find where the numerator is negative. And we find that for n greater than 100 over log 1.01, the numerator is negative. And so the terms are eventually decreasing. It's important not to get too lost in the algebra. The critical question isn't so much when do we become eventually, but rather do we become eventually. And in some cases it's possible to answer the question of whether we become eventually without determining when we become. For example, here's a series and we'd like to know whether this becomes eventually alternating. Well, maybe the universe is kind to us and maybe I am a kind and gentle math teacher and I've given you an alternating series. Don't count on it. But hope springs eternal, so we might try to find the first few terms. And these terms do not alternate between plus and minus, so this is not an alternating series. Now let's pause a moment and think about this. Ordinarily, when we see minus 1 to power n, we expect an alternating series because minus 1 to n will be positive or negative depending on whether n is even or odd. So why isn't that working in this case? The problem is we're getting interference from the value of sine, which could be positive or negative. However, we might consider the following. As n goes to infinity, 7000 over n goes to 0 from above, and consequently sine of 7000 over n also goes to 0 from above. And what this means is that once n gets large enough, sine of 7000 over n will be positive, and the sine of the terms of the series will be determined entirely by minus 1 to power n. So eventually, the trigonometric factor is positive, and the sine, S-I-G-N, of the terms of the series, is going to be determined entirely by the minus 1 to power n, which will alternate, and so this series is eventually alternating, even though without more work we won't know when that happens. So let's put this all together. Suppose the terms of a series are eventually, then the tail of the series is always. Since convergence only depends on the tail, then any series that is eventually can be treated as a series that is always. And this leads us to one very important test, which unfortunately is actually a test for divergence. And this is known as the nth term test. Suppose the limit as n goes to infinity of An is not equal to 0, then the infinite series whose terms are An diverges. And importantly, if the limit as n goes to infinity of An is 0, the test is inconclusive. We don't know whether the series converges or diverges. We have to do more testing. An important thing to recognize is that our nth term test can give us an inconclusive result. It can leave us not knowing whether a series converges or diverges. And in fact, this is a general feature of convergence test. All convergence tests should be used with caution. The nth term test will only tell you when a series diverges. And so there are three important things to remember about the nth term test. First, the nth term test does not tell you whether a series converges. It can only tell you a series diverges. Second, the nth term test does not tell you whether a series converges. It can only tell you a series diverges. And third, and most importantly, the nth term test does not tell you whether a series converges. It can only tell you a series diverges. So let's determine whether this series, as n goes from 0 to infinity of sine n, converges or diverges. So we'll pull in our nth term test and see if we can find the limit as n goes to infinity of sine of n. And we observe that as n goes to infinity, sine n does not approach any fixed value. So the limit as n goes to infinity of sine of n does not exist. And our nth term test is fairly specific. If the limit is not equal to 0, then the series diverges. Does not exist is different from 0, so the series diverges. How about the series whose terms are 1 over n? Again, as an infinite series, the convergence or divergence of the series does not depend on where we start, but rather on the tail of the series. So let's apply the nth term test. We find the limit as n goes to infinity of 1 over n, which is 0, but we should remember the three things about the nth term test. The nth term test does not tell you whether a series converges. It can only tell you a series diverges. And in this case, since the limit is 0, the test is inconclusive. We don't know whether this series converges or diverges. Put together, this leads to the following approach to finding the convergence or divergence of a series. First, if the nth term test fails, the series is divergent. Again, it's important to remember the nth term test does not tell you whether a series converges. It can only tell you whether a series converges. If the limit of the nth term is 0, the nth term test is inconclusive, which means we have to do a little bit more work. So our next step is to determine if the series is eventually decreasing. And if it is, we can use a convergence test. We'll introduce several of those in the next few videos. Unfortunately, if the series is not eventually decreasing, well, cry. And that's because determining convergence or divergence will be extremely difficult.