 Hello and welcome to another screencast on Taylor series. This one picks up where the last one left off. So if you have not watched screencast 8.5.4, this one's not gonna make a lot of sense to you. So go ahead and stop this one, go back and watch the next one. And trust me, these need to be watched in order. Okay, so where we left off the last time then is we decided that the Taylor series for the natural log of X is going to follow a certain pattern. Okay, so when K is one, that's called an order one Taylor series, we end up getting the, actually the line as it turns out, of X minus one. Okay, now our Taylor series is centered at one. So you'll notice right around one, our approximation is actually really good, right? My red line comes really, really close to my function L of X. It's only until we get about, I don't know, maybe 0.4, 0.6 away, do these two pieces start to separate. Okay, let's take a look when K is two. So when K is two, I've got this Taylor series, again following the pattern you saw in the last video, centered at one. Okay, so looking here, I'm still pretty close to when I go 0.4 away. When I'm 0.6 away, I see some separation on this side, but on the right side, I don't see any separation really until we're about 0.8 away. Okay, let's look at the next K. So when K is three, I end up with this polynomial. So you'll notice it's getting bigger, but it's following the same pattern that you saw in the last video. Again, we're centered at one. So looking to the left, I'm pretty close until about 0.2. And to the right, I'm really close until, maybe somewhere between 1.8 and two. So again, we're getting good approximations here to our function with this particular polynomial. Okay, order four. So this one I think is looking the best one yet. Here we go, centered at one. So if I go out to the right until again, maybe about two, somewhere between two and 2.2, it starts to, you know, it's called diverging. They start to go away from each other. And then over here on the right, I'm really, really close at 0.2. And then things start to fall off here when I get close to zero, which makes total sense, because the natural log function has an asymptote at zero. Okay, so things are definitely gonna be shooting down towards negative infinity right there. We're a polynomial that's not gonna happen, okay? So all of this leads us then to the natural question, well, how good is our approximation? How far can we go away from this particular value we're centered at and still have a good approximation, still have convergence, okay? So that leads us to our question. Given the Taylor series for the function we've been working with, natural log of x, centered at x equals one, okay, is defined by, so this is what you guys, what we figured out in the last video. We showed you how to get that. So our series is k goes from one to infinity, negative one to the k plus one divided by k times x minus one to the k. Well, that's a k's in there. All right, so anyway, so that is our Taylor series. Now, I want you to use the ratio test to find the radius and the interval of convergence. Okay, so the radius is gonna tell us how far away we can go from that center point of x equals one and still have our approximation be good. Okay, then the interval, what we're gonna do then is we're gonna test our endpoints and we're gonna see, does it converge at our endpoints or not? Okay, so let's start with our ratio test. So hopefully you remember that the ratio test says the limit as k goes to infinity, then here we wanna use our function, oops, I'll just write this out, the absolute value of a sub k plus one divided by the absolute value of a sub k. Okay, so our function here is our a sub k. So then how do we get a sub k plus one? Well, that just means wherever we see a k, we're gonna replace it with k plus one. And we're gonna throw all this in absolute values and we're gonna make a big old mess with this stuff. Let me tell you. Okay, so we've got negative one to the k plus one plus one, okay, k plus two, divided by k plus one times x minus one to the k plus one. Okay, that's gonna be divided by a sub k, so it's just this whole thing rewritten inside of those absolute values. Okay, now our job is to simplify this, so let's throw some algebra at it first before we even bother with this limit. But I'm gonna rewrite it every time, of course, so I don't lose it. Okay, so looking at these absolute values, well, what happens to negative one to the k plus two in absolute value? Well, that's just gonna give us one, right? Because negative one to the k plus two is gonna oscillate between negative one and one, but that's pretty much just gonna drop out. Okay, bam, that goes away. Now, what about the k plus one? What happens to that in absolute value? Well, remember, k starts at one and goes to infinity, so the absolute value isn't gonna affect it at all. Okay, same thing with this exponent up here, the absolute value's not gonna affect that, so this whole numerator then simplifies into the absolute value of x minus one, because that obviously will get affected, right? X can be anything, so x could be, you know, negative two or something, so then that absolute value is gonna make a difference. So that's gonna be to the k plus one all over k plus one. Similar argument with our denominator, so this piece wipes out, the absolute value slides inside of that, oopsie, exponent, and we've got that. Okay, we're getting there. So now I see a fraction over a fraction, so let's go ahead and multiply by the reciprocal. Let me actually go down to the next line here, so I'm not squeezing stuff in too much. So I've got the absolute value of x minus one to the k plus one all over k plus one times, I'm gonna go ahead and flip that denominator over, so I've got k over x minus one to the k. All right, hopefully you guys are buying this algebra so far. Okay, so now I'm gonna kinda, let's break these two things apart, so let's get the k's over here together and all the x minus one and absolute values together. So we have the limit, k approaches infinity, k over k plus one, absolute value x minus one to the k plus one, all over absolute value of x minus one to the k. All right, now where do we start with this? Well, I'm gonna take a look at the second piece here. So looking at this piece, hopefully you guys can see we actually have the same base, as ugly as it is, right? So then when we divide, remember we can subtract our exponents. Lovely, look at what's gonna happen then. So we're gonna have k plus one and then minus k, so that's just gonna give us the absolute value of x minus one, woohoo, all right, that's nice. Okay, what about this other piece here? So let me highlight that one in a different color. So what happens to k over k plus one as k goes to infinity? Well, it's infinity over infinity. So that means we can use L'Hopital's, right, which we're gonna end up with doing our derivatives one over one, so that's just gonna give us one. Ooh, that's nice. Okay, now thinking back to our ratio test, we know that whatever we have left here converges when this stuff is going to be less than one. Okay, now this one times all this stuff doesn't really matter, so this is really the part that we are focused in on. Okay, so this tells us then that our radius of convergence is one. Okay, let me draw out a number line. So I'm centered at one. I'm gonna go one to the right and I'm gonna go one to the left. And pretend like those are actually the same length. Sorry about that. So if I go one to the left, that's gonna put me at zero. If I go one to the right, that's gonna put me at two. Okay, so this is my possible interval of convergence going from zero to two. So now I need to test these endpoints. All right, so when x equals zero, let's see what we get. So remember, our series was negative one of the k plus one all over k. Now I'm gonna plug in my values for x here, okay? So that gives me the sum k goes from one to infinity, negative one to the k plus one all over k, times, then that's gonna end up giving me zero minus one to the k. All righty, so that gives us, doing a little bit of simplification here, this is really negative one to the k, right? So then we can combine these negative one bases. So we're gonna end up with k goes from one to infinity, negative one to the two k plus one all over k. All right, so does this thing converge or diverge? I'm not totally convinced yet. So let me use another property of exponents and break up this numerator. So I'm gonna rewrite this as negative one to the two k, times negative one to the one all over k. All right, well, negative one to the two k, that means it's gonna be negative one to an even power. So this piece is actually just gonna give us one. So we really then have the series, k goes from one to infinity of negative one over k. Well, this is the opposite of the harmonic series and you guys know harmonic series diverges. Okay, so that means we are not gonna include zero. So when I go to write my interval, which I'll do down here at the bottom, we're not gonna include zero. And again, that makes perfect sense because it has an asymptote there at the natural log dose. And remember, that's where this whole Taylor series stuff came from. Okay, let's look at two and let's see what happens at two. K goes from one to infinity. So these problems really have a lot to them. You know, there's a lot of stuff involved. But it's all bringing pieces together that we've done before, so that's always good. Okay, well, two minus one is just one of the k. One of the k is anything, or is just one. One of the anything I should say is one. All right, so that's just gonna give us negative one to the k plus one all over k. This series alternates, right? So by the alternating series test, this one converges. So we will include zero, or include two. I forgot what number I was doing. All right, so my interval of convergence then is gonna look like the open parentheses. So zero to two, and then with a closed bracket. So this tells me then that my series is going to be very close to my original function. Let me go back and look at this. So here, again, this makes perfect sense. At zero, we have this total separation but over here at two, we're pretty darn close. All right, thank you for watching.