Which series converge and which diverge:1.2.Vn3, 7S n=i...

Which series
converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=1}^{\infty}\left(\frac{n}{n+1}-\frac{n+2}{n+3}\right)$$

Want to determine whether this Siri's will converge or diverge. Now you might look at this and say, Oh, looks like integral tests, cause we could let you be the square would have been Plus, one derivative of that would give us a way to cancel out that one of the square root of n plus one. But we come up with an issue so eventually that's going to come down to the integral of something, one over the natural log of you after you do the substitution. So we should still have, like, some constants being multiple assault to say some constancy doesn't really matter. And this is a non elementary integral. So if you're allowed to actually like, say that this is just a non elementary integral from here, it's pretty straightforward to say that this diverges. But in the case of you not being able to know how to do one over the natural log of you, we can actually use the direct comparison test because one thing that we actually have is that if we look at the natural log of the square root of N plus one, this is going to be less than or equal to just square root of n plus one, especially for n greater than or equal to three. And if we were to reciprocate each side So divide over, we would end up with this here so we could actually use the direct comparison test to rewrite this in the following way. So this is going to be greater than or equal to the sum from in. Does he go to three to infinity of seven over the square root of in plus one times the square root have been plus one. Those square roots are gonna cats that with each other, I'm gonna factor up that seven as well. So this is gonna be the some from n is equal to three to infinity of one over in plus one and we'll weaken. Compare this again to one over, in So there's gonna be the some from three. Where is he going? To three to infinity Time seven. And actually, we can go ahead and get rid of that seven. Since multiplying this, uh, larger number will always be larger. So I would just be one over it. And we know that this di vergis by well, is the harmonic series, so it should diverge, but also by P. Siri's P series, where P is equal one, and that's greater than or equal to one. And so this is going to go to infinity, which implies that our original Siris in is he was three to infinity of seven over in plus one square rooted natural log of the square root of and plus one is going to be greater than or equal to infinity, which implies it die Burgess.

Were given following Siri's and you're asked, Figure out whether or not the series converges or diverges. And if it converges your ass to find this up, do you start off for breaking down? Using partial fraction decomposition Get treated on over 32 on minus 1/3 21 Next thing you knew start writing out the 1st 2 terms of the geometric. Siri's deceiving and find some sort of product, So Starsia n equals zero. So you get one plus one plus 2/3 plus 1/3 plus four nights plus one night said, uh and this is when you begin to find a pattern. It smells that used 101st. The first term of the first part of each turn is being multiplied. But your shirt, the second part of each term, is being multiplied by 1/3. And you know that hiss. The absolute values are which is amounts for the most part is off someone this convergence and is this true? Well, in this case, this is our This is our and they're both true. So you know that this is true and you know about this series converges so the next thing you want to do is he wants to figure out what the some of the serious is. Because now that you know, that converges the question. I asked you find a song who used a formula. Okay. Over one. Mice are. And because our Children you do minus a over one month, sir. Um and that's gonna get you. You go tails who excuse out of here is equal to. So you're starting values one 1/1, minus 2/3 minus one over one. Nice 1/3. Which gets you when mice 2/3 is 1/3 and divide by. That is just one socialist you get when mice wondered which is one over wondered, minus one over to third, which is equal to three minus 3/2, which is simply equal to 3/2 so that you know that the serious coverages in the summer Siri's simply 3/2

Hello, everybody. And welcome back. This is Kevin Shock with New Murad. Let's consider the infinite Siri's from n equals zero to infinity of the square root of two to the end. Power. Well, what we know we're gonna be doing here is we're gonna be doing to hopefully make sure they get a nice with years of all races up with. We're gonna start with value of two, and we're going be raising that to the and over to power each time. So for the zeroth power, that's gonna be to the zero or two. And then we're going to be adding that to to to the 1/2, adding that to to to the to over to 3/2, etcetera. Now I noticed that this value is going to be one. This value is going to be the square root of two. This value is going to be too. Oh, can we see what's occurring here? Each of these times that I add, I'm adding mawr and I'm adding mawr. So I definitely don't have ah situation where the limit as M goes to infinity of my partial summation up to em is definitely not going to be a zero. In fact, it's gonna be equal to infinities. We're gonna say this is equal to infinity, which is most definitely not zero. And therefore, this implies that our Siri's is going to diverge. So we can say that are full summation with no subscript, which just means are infinite summation eyes going to diverge or, in other words, is going to go up to infinity. Let's actually clarify this just a bit. We're gonna say it goes up to not It is because it can't equal. So are are summation goes up to infinity approaches infinity.

You were given a falling seriously ask to determine whether or not the series convergence or divergence and convergence you're asked to find the some of the Siri's. So start off with You can break the sound, given Siri's and to tooted on Over 49 plus three two and over Fortunate. So you can start by writing out the first returns of seriousness, even find patters so you will get 1/2 plus 3/4 plus 1/4 who has 9/16 plus 1/8 plus 27/64 etcetera. You will begin to those of patterns us that each time the first part of each term is being multiplied by 1/2 in the second part during multiplied right three force. You know that the Siri's converges if the absolute value of our is lost in one where ours amount of dust being most part each times in this case, everyone half and three force in this case, absolute value of our is lost on one. So you know this is true. And so you know that the Siri's converges now because, you know, converges you're not asked to find a song. So you can simply use a Formula 8/1. Rights are plus fate over one. Rights are which is equal to where a is the first time seriously would be 1/2 over a one minus. The masking wants White each time. 1/2 plus three force over one My story force, which is equal to 1/2 but about 1/2 plus three forced right by 1/4. We're just simply equal to one plus three, which is you go to four so that some of Siri's is for so you know that his face covered is you know, some of Siri's. It's for