Alternating series test for convergence
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Alternating series test
- [Narrator] Let's now expose ourselves to another test of convergence, and that's the Alternating Series Test. 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. 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 let's say A sub N, I can write. 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 Ns we care about. So for all of these integer Ns greater than or equal to K. If all of these things, 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. Number two, B sub N is a decreasing sequence. Decreasing... Decreasing sequence. Then that lets us know that the original infinite series, the original infinite series, is going to converge. So this might seem a little bit abstract right now. Let's actually show, let's use this with an actual series to make it a little bit more, a little bit more concrete. Let's say that I had the series, let's say I had the series from N equals one to infinity of negative one to the N over N. We could write it out just to make this series a little bit more concrete. When N is equal to one, this is gonna 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. When N is equal to one, this is gonna be negative one squared over one which is gonna be one. Then when N is two, it's gonna be negative one to the third power which is gonna be negative one half. So it's minus one half plus one third minus one fourth plus minus and it keeps going on and 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 negative, 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 is, which we can then say this thing right over here could be our B sub N. This right over here is our B sub N. We can verify that our B sub N is going to be greater than or equal to zero for all the Ns we care about. So our B sub N is equal to one over N. Clearly this is gonna be greater than or equal to zero for any, for any positive N. Now what's the limit? As B sub N, What's the limit of B sub N as N approaches infinity? The limit of, let me just write one over N, one over N, as N approaches infinity is going to be equal to zero. So we satisfied the first constraint. Then this is clearly a decreasing sequence as N increases the denominators are going to increase. With a larger denominator, you're going to have a lower value. We can also say one over N is a decreasing, decreasing sequence for the Ns that we care about. So this satifies, this is satisfied as well. Based on that, this thing is always, 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 originial series actually converges. So N equals 1 to infinity of negative one to the N plus 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 just have the Harmonic Series, and that one didn't converge. But this one did, putting these negatives here do the trick. Actually we can prove this one over here converges using other techniques. 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 about like that Divergence Test, but remember the Divergence Test is really, is only useful if you want to show something diverges. If the limit of, 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. 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.
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