B) convergence Test of 1 0 0 1 7 methoxdl liz + 7 1 1 Sor " " 4 suitable 2 4 0 Following x Ij J guaranteed8 Find the 8 lactorization ol the 3 1...

Test the series for convergence or divergence.
$ -\frac {2}{5} + \frac {4}{6} - \frac {6}{7} + \frac {8}{8} - \frac {10}{9} + \cdot \cdot \cdot $

Asian seriously in expansion form. Working in the ministry out of far no. Five out of five, minus seven out of six last night out of seven, minus 11 under paint and so on. And we can re baptism in the series from where we can, uh, minus one of the end. And then we have the two endless won the fighting by the two. And, uh and it goes to one here and then with the energy members to here on the Angus from one to infinity. And now yeah, notice that we can use that. I wasn't danced in here when we were computer limit. After I'm sort of in now and goes to infinity, and then we get a limit off. And with infinite day, I'm going to embrace Gone over two. And this one will be on leader in now. Ah, and plus three coffee. And then you have the embers three here and now we see that energy infinity. You can compute this limit. Yeah, by a factor that in outside in somehow the Jew plus one off him. You have anything in something one plus two other and and we suit and consider end with the end and it's Anna goes to infinity this time goes to zero. That's where this one goes to zero. And then we get this limited coaching to over one and culture to and because it was not in court, you zero. Therefore, we conclude that this reasonably be divergent.

This is either conditionally or absolutely conversions. So take a look at this year. Let's start out with the check and see if it's absolutely convergent here. We have the house the value of negative 12 K over to UK here. The moment passed. So we take this key approaches infinity here. Actually know before that. Let's just take a look and see what happens. Damn civilian here. So then we have here that this is equivalent to 1/2 K. So summation of chemicals to infinity here. And we know that that's so this is harmonic series. All right. And so therefore this diverges Because it's like 1/2 times Submission of one hurricane. You know that that diverges there for sure. So then we have kicked. So our next part if we take a look at negative one to the K over to UK here. So as it is then this is by alternating series We see that so a one as So our first term is equal to 1/4. Our second term is over six. And so then see that a one is greater than a two and so on. So just keep going Is greater than the next one which is one of great. So then he by alternate series. We know that this here. So the limit has cave purchase affinity of one of her two K goes to zero. So I was training serious this come purchase. So we would say that this series converges conditionally only

Okay. In the successes, we have to determine if he Siri's converges off my purchase. One interesting fact, please, that we are asked not to check seeds and sickle to want to infinity. We have to check for any court straight infinity in order to lead. We're gonna work with the serial one over and Q But we're gonna start from three T and Seed and we know that the serious converges because he's happy theory with Peak waas three. And because of the remark in the page 602 that tell us that if the whole Siri's converges when we shift it, he's also gonna call Burge. So we're gonna use the Lei Me comparison kiss. We're gonna calculate any meat going in goes to infinity and winter. We're gonna divide the general terms. This is the same that the limit gun and goes to infinity open you minus eight over in Cuba. Now we can simply file live it and we get one miner's eight over Thank you. And this term is gonna go to Syria when and ghost infinity and therefore the diva's one. Since one is a positive number. We know that both here is called Bertram or Siri's Diverge. And since the series convergent the original one he sold, say, for some bridge, yeah.

Let's see whether the Siri's converges are emerges. Now we see that it's alternating. However we see that the numerator increases no sloppy there, let me backtrack. Increases by two, whereas the denominator increases on ly by one so we can actually rewrite this Erie's Yeah, let's say up top. First of all, we should have this negative one to the end power and then up top. We should have two in and then in the denominator, we're adding one each time. But we're starting with an equals one. So four plus end. So here, let's just look at this term here are in. And then let's just go find B End to just be to win over four percent. Now I know that the limit of the end is just equal to two, which is non zero. So this implies that the limit of A M is undefined sense. As n gets really, really large way want will keep all supplying by negative one. But this fraction over here is getting closer to to sew in the limit. A N is getting very close to negative two two negative to two and so on. It's all the limit just will not exist. Therefore, our Siri's we're given above diverges bye, the diversions test, and that's our final answer